A MODERN INTRODUCTION TO PARTICLE PHYSICS Second Edition
FAYYAZUDDIN & RIAZUDDIN
World Scientific Publishing
A MODERN INT~ODUCTI~N TO
PARTICLE PHYSICS Second Edition
A MODERN INTRODUCTION TO
PARTICLE PHYSICS Second Edition
FAYYAZUDDIN & RIAZUDDIN National Center for Physics Quaid-e-ham University Pakisfan
World Scientific Singapore New Jersey. London Hong Kong
Published by
World Scientific PublishingCo. Pte. Ltd. P 0 Box 128, Farrer Road, Singapore 912805 USA ofice: Suite lB, 1060 Main Street, River Edge, NJ 07661 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-PublicationData Fayyazuddin, 1930A modem introduction to particle physics I Fayyazuddin, Riazuddin -- 2nd ed. p. cm. Includes bibliographicalreferences and index. ISBN 9810238762 (alk. paper) ISBN 9810238770 (pbk) 1. Particles (Nuclear physics) I. Riazuddin. 11. Title. QC793.2 .F39 2000 539.7'2--dc21
00-035183
British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.
Copyright 0 2000 by World Scientific Publishing Co. Pte. Ltd. AI1 rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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Thou seest not in the creation of Allmerciful any imperfection. Return thy gaze; seest thou any fissure? Then return thy gaze again, and again and thy gaze comes back to thee dazzled, aweary Koran, The Kingdom LXVII
Preface to the first edition Particle physics has been one of the frontiers of science since J. J. Thompson’s discovery of the electron about one hundred years ago. Since then physicists have been concerned with (i) attempts to discover the ultimate constituents of matter, (ii) the fundamental forces through which the fundamental constituents interact, and (iii) seeking a unification of the fundamental forces. At the present level of experimental resolution, the smallest units of matter appear to be leptons and quarks, which are spin 1/2 fermions. Hadrons (particles which feel the strong force) are composed of quarks. The evidence for this comes from the observed spectrum and static properties of hadrons and from high energy lepton-hadron scattering experiments involving large momentum transfers, which ”prove” the actual existence of quarks within hadrons. As originally formulated, the quark model needed three flavors of quarks, up ( U ) , down ( d ) and strangeness (s) not just U and d. The discoveries of the tau leptons and more flavors [charm (c) and bottom ( b ) ] were to some extent welcomed and to some extent appeared to be there for no apparent reason since elementary building blocks of an atom are just U and d quarks and electrons. A charm quark was predicted to exist to remove all phenomenological obstacles to a proper and an elegant gauge theory of weak interaction. Without it, nonexistence of strangeness-changing neutral current posed a puzzle. This also restored the quark-lepton symmetry: for each pair of leptons of charges 0 and -1 there is a quark pair of charges 2/3 and -1/3. The existence of 7-leptons ix
X
Preface to the first edition
and discovery of the b quark (charge -1/3) demand the existence of another quark (charge 2/3), called the top quark, to again restore the quark-lepton symmetry. Indeed, six quark flavors have been proposed to incorporate violation of CP invariance in weak interaction. Quarks also have a hidden three valued degree of freedom known as color: each quark flavor comes in three colors. The antisymmetry of threequark wave function of a baryon [e.g. proton] is attributed to color degree of freedom. The three number of colors also manifest themselves in no decay and in the annihilation of lepton-antilepton into hadrons. We have encountered the following types of charges: gravitational, namely, mass, electric, flavor and color. The fundamental forces through which elementary fermions interact are then simply the forces of attraction or repulsion between these charges. The unification of forces is then sought by searching for a single entity of which the various charges are components in the sense that they can be transformed into one and another. 'In other words, they form generators of a gauge group G which is taken tjo be local so that a definite form of interaction between vector fields (which must exist and belong to the adjoint representation of G) and elementary fermions (which belong to the fundamental or trivial representation of G) is generated with a universal coupling constant. In this respect non-Abelian gauge field theories [Yang-Mills type] have played a major role. Here the field itself is a carrier of "charge" so that there are direct interactions between the field quanta. Let 11s first discuss the strong quark interactions. The local gauge group is SUc(3) generated by three color charges, the field quanta are eight massless spin 1 color carrying gluons. The theory of quark interactions arising from the exchange of gluons is called quantum chromodynamics (QCD). The most, striking physical properties of QCD are (i) t,he concept of a "running coupling constant a3( q 2 ) " ,depending on the amount, of momentum transfer q2. It goes to zero for high q2 leading to asymptotic freedom and becomes large for low q2, (ii) confinement of quarks and gluons in
Preface to the first edition
xi
a hadron so that only color singlets can be produced and observed. Only the property (i) has a rigorous theoretical basis while the property (ii) finds support from hadron spectroscopy and lattice gauge simulations. Weak and electromagnetic interactions result from a gauge group acting upon flavors. It is SUL(2) x U( 1) and is spontaneously broken rather than exact as was SUc(3). The electroweak theory, together with the quark hypothesis and QCD, form the basis for the so called "Standard Model" of elementary particles. There have been many quantitative confirmations of the predictions of the standard model: existence of neutral weak current mediated by Zo, discovery of weak vector bosons W*, Zo at the predicated masses, precision determinations of electroweak parameters and coupling constants (e.g. sin2Bw which comes out to be the same in all experiments) leading t,o one loop verification of the theory and providing constraints on the top quark and Higgs masses. Similarly there have been tests of QCD, verifying the running of the coupling constant a3(q2), q2dependence of structure functions in deep-inelastic lepton-nucleon scattering. Other evidences come from hadron spectroscopy and from high energy processes in which gluons p,lay an essential role. In spite of the above successes, many questions remain: replication of families and how many quarks and leptons are there? QCD does not throw any light on how many quark flavors there should be? Origin of fermion masses, which appear as free parameters since Higgs couplings with fermions contain as many arbitrary coupling constants as there are masses, is another unanswered question. Origin of CP violation at more fundamental level, rigorous basis of confinement and hadronization of quarks are other questions which await answers. Top quark and Higgs boson are still to be discovered. Symmetry principles have played an important part in our understanding of particle physics. Thus Chapters 2-6 discuss global symmetries and flavor or classifications symmetries like SU(2) and SU(3) and quark model. Chapter 5 provides the necessary group
xii
Preface to the first edition
theory and consequences of flavor SU(3). Chapters 2-6 together with Chapters 9, 10 and 11 on neutrino, weak interactions, properties of weak hadronic currents and chiral symmetry comprise mainly what, is called old particle physics but, include some new topics like neutrino oscillations and solar neutrino problem. These Chapters are included to provide necessary background to new particle physics, comprising mainly the standard model as defined above. The rest of the book is devoted to the standard model and the topics mentioned in paras 2-7 of the preface. Recently there has been an interface of particle physics with cosmology, providing not only an understanding of the history of very early universe but, also shedding some light on questions such as dark matter and open or closed universe. Chapter 16 of the book is devoted to this interface. Particle physics forms an essential part, of physics curriciilum. This book can be used as a text, book, but it, may also be useful for people working in the field. The book is so designed as to form one semester course for senior undergraduates (with suitable selection of the material) and one semester course for graduate students. Formal quantum field theory is not, used; only a knowledge of non-relativistic quantum mechanics is required for some parts of the book. But, for the remaining parts, the knowledge of relativistic quantiini mechanics is essential. The familiarity with quantum field theory is an advant,age and for this purpose two Appendicess which summarize the Feynman rules and renormalization group techniques, are added. Initial incentive for t,his book came from the lectures which we have given at, various places: Quaid-e- Azam University, Islamabad, Daresbury Nuclear Physics Laboratory (R), the University of Iowa (R), King Fahd University of Petroleum and Minerals, Dhahran (R) and King Abdulaziz University, Jeddah (F). We have not prepared a bibliography of the original papers underlying the developments discussed in the book. Remedy for this can be found in the recent review articles and books listed at, the end of each Chapter.
Preface to the first edition
xiii
We wish to express our deep sense of appreciation t,o Dr. Ahmed Ali for critically reading the manuscript,, for making many useful suggestions and for his help to update the data. We also wish to express our deep thanks to a colleague Mr. El hassan El aaud and a graduate student Mr. F. M. Al-Shamali [of one of us (R)], who drew diagrams and in general assisted in producing the final manuscript. In addition, the typing help provided by Mr. Mohammad Junaid at Research Institute of King Fahd University of Petroleum and Minerals was indispensable in getting the job done. Finally we wish to acknowledge the support of King Fahd University of Petroleum and Minerals for this project under Project No. PH/Particle/l23. We also take this opportunity to express our deep sense of gratitude to Prof. Abdus Salam, who first introduced us to this subject and for his encouragement throughout oiir work in this field. Fayyazuddin Riazuddin March 4, 1992
Contents 1 Introduction 1 1.1 Fundamental Force . . . . . . . . . . . . . . . . . . 1 1.2 Classification of Matter: Leptons and Quarks . . . 8 1.3 Strong Color Charges . . . . . . . . . . . . . . . . . 10 1.4 Fundamental Role of “Charges” and the Standard Model of Electroweak Unification and Strong Force 12 1.5 Strong Quark-Quark Force . . . . . . . . . . . . . . 19 1.6 Grand Unification . . . . . . . . . . . . . . . . . . . 21 1.7 Units and Notation . . . . . . . . . . . . . . . . . . 24 1.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . 26
2 Scattering and Particle Interaction 2.1 Kinematics of a Scattering Process . . . . . . . . . 2.2 Interaction Picture . . . . . . . . . . . . . . . . . . 2.3 Scattering Matrix (S-Matrix) . . . . . . . . . . . . 2.4 Phasespace . . . . . . . . . . . . . . . . . . . . . . 2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Two-body scattering . . . . . . . . . . . . . 2.5.2 Three-body decay . . . . . . . . . . . . . . . 2.6 Electromagnetic Interaction . . . . . . . . . . . . . . 2.7 Weak Interaction . . . . . . . . . . . . . . . . . . . 2.8 Hadronic Cross-section . . . . . . . . . . . . . . . . 2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Bibliography . . . . . . . . . . . . . . . . . . . . . . xvii
27 27 31 33 38 41 41 43 52 57 60 61 63
xviii
Contents
3 Space-Time Symmetries 3.1 Invariance Principle . . . . . . . . . . . . . . . . . . 3.2 Parity . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Intrinsic Parity . . . . . . . . . . . . . . . . . . . . 3.4 Parity Constraints on S-Matrix for Hadronic Reactions 3.4.1 Scattering of spin 0 particles on spin particles 3.4.2 Decay of a spin O+ particle into three spinless particles each having odd parity . . . . . . . . 3.5 Time Reversal . . . . . . . . . . . . . . . . . . . . . 3.6 Applications . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Detailed balance principle . . . . . . . . . . . 3.7 Unitarity Constraints . . . . . . . . . . . . . . . . . 3.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Bibliography . . . . . . . . . . . . . . . . . . . . . .
65 65 67 69 73 73
4 Internal Symmetries 4.1 Selection Rules and Globally Conserved Quantum Numbers . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Isospin . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Electromagnetic interaction and isospin . . . 4.2.2 Weak interaction and isospin . . . . . . . . . 4.3 Resonance Production . . . . . . . . . . . . . . . . 4.4 Charge Conjugation . . . . . . . . . . . . . . . . . . 4.5 G-Parity . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Bibliography . . . . . . . . . . . . . . . . . . . . . .
97
5 Unitary Groups and SU(3) 5.1 Unitary Groups and SU(3) . . . . . . . . . . . . . . 5.2 Particle Representations in Flavor SU(3) . . . . . . 5.3 U-Spin . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Irreducible Representations of SU(3) . . . . . . . . 5.5 SU(N) . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Applications of Flavor SU(3) . . . . . . . . . . . . . 5.7 Mass Splitting in Flavor SU(3) . . . . . . . . . . . .
75 76 79 79 80 91 95
97 106 110 111 111 120 125 127 129
131 131 137
151 151 159 167 170
xix
Contents
5.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . 6 SU(6) and Quark Model 6.1 SU(6) . . . . . . . . . . . . . . . . . 6.2 Magnetic Moments of Baryons . . . 6.3 Radiative Decays of Vector Mesons 6.4 Problems . . . . . . . . . . . . . . . . 6.5 Bibliography . . . . . . . . . . . . . .
178 183
185 ........ 185 . . . . . . . . . 192 . . . . . . . . . 200 ........ 209 ........ 211
7 Color. Gauge Principle and Quantum Chromodynamics 213 7.1 Evidence for Color . . . . . . . . . . . . . . . . . . 213 218 7.2 Gauge Principle . . . . . . . . . . . . . . . . . . . . 7.2.1 Aharanov and Bohm experiment . . . . . . 220 7.2.2 Gauge principle for relativistic quantum mechanics . . . . . . . . . . . . . . . . . . . . . 223 7.3 Quantum Chromodynamics (QCD) . . . . . . . . . 225 7.3.1 Conserved current. . . . . . . . . . . . . . . . 228 7.3.2 Experimental determinations of aS(q2)and asymptotic freedom of QCD . . . . . . . . . 231 7.4 Hadron Spectroscopy . . . . . . . . . . . . . . . . . 237 7.4.1 One gluon exchange potential . . . . . . . . 237 7.4.2 Long range QCD motivated potential . . . . 239 7.4.3 Spin-spin interaction . . . . . . . . . . . . . 243 7.5 The Mass Spectrum . . . . . . . . . . . . . . . . . . 244 7.5.1 Meson mass spectrum . . . . . . . . . . . . 246 7.5.2 Baryon mass spectrum . . . . . . . . . . . . 250 7.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . 255 8 Heavy Flavors 8.1 Discovery of Charm . . . . . . . . . . . . . . . . . . 8.1.1 Isospin . . . . . . . . . . . . . . . . . . . . . 8.1.2 SU (3) classification . . . . . . . . . . . . . . 8.2 Charm . . . . . . . . . . . . . . . . . . . . . . . . .
259 259 261 261 262
Preface to the second edition Our aim in producing this new edition is to bring the book up to date and as such many chapters have been throughly revised. In particular, the chapters on Neutrino Physics, Particle Mixing and CP-Violation and Weak Decays of Heavy Flavors have been mostly rewritten incorporating new material and new data. The heavy quark effective field theory has been included and a brief introductory section on supersymmetry and strings has been added. We wish to thank Ansar Fayyazuddin for writing this section. A number of typographical errors have been corrected. Another change is that we have adopted a metric and notation for gamma matrices commonly used.* Finally we wish to thank Mr. Amjad Hussain Gilani and Dr. Muhammad Nisar who did an excellent, job in typing t,he manuscript; without their help it was difficult to piit the manuscript in final shape. Fayyazuddin Riazuddin Jan. 21, 2000
*see for example, J. D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill Book Co., New York (1965).
xv
Chapter 1 INTRODUCTION 1.1
Fundamental Force
Particle physics is concerned with the fundamental constituents of matter and the fundamental “forces” through which the fundamental constituents interact among themselves. Until about 1932, only four particles, namely the proton ( p ) , the neutron ( n ) ,the electron (e) and the neutrino (v) were regarded as the ultimate constituents of matter. Of these four particles, two, the proton and the electron are electrically charged. The other two are electrically neutral. The neutron and proton form atomic nuclei, the electron and nucleus form atoms while the neutrino comes out in radioactivity, i.e. the neutron decays into a proton, an electron and a neutrino. Each of these particles, called a fermion, spins and exists in two spin (or polarization) states called left-handed (i.e. appears to be spinning clockwise as viewed by an observer that it is approaching) and right-handed (i.e. spinning anti-clockwise) spin states. One may add a fifth particle, the photon to this list. The photon is a quantum of electromagnetic field. It is a boson and carries spin 1, is electrically neutral and has zero mass, due to which it has only two spin directions or it has only transverse polarization. It is a mediator of electromagnetic force. A general feature of quantum field theory is that each particle has its own antiparticle with opposite charge and magnetic moment, but with same mass and spin. Accordingly we have four antiparticles viz., the antiproton ( p ) , the positron ( e + ) , the antineutron (Fi) 1
2
Introduction
and antineutrino ( V ) . The four particles experience four types of forces:
i. The Gravitational Force This is a force of attraction between two particles and is proportional to their gravitational charges, namely their masses. It is a long range force, controls the motion of planets and galaxies, governs the law of falling bodies, and determines the overall character of our Universe. The gravitational potential energy between two protons is given by the Newton’s Law:
where GN is the Newton’s gravitational constant:
GN = 4.17 x 10-5GeV-cm/gm2 = 0.67 x 10-38GeV-2.
For proton
mp
M
lGeV, r = 10-13cm M 5GeV-’,
V
M
10-39GeV.
(1.3)
Thus we see that on microscopic scale, the gravitational potential energy is negligible. But we note that
fi - 1-
m
1 - M 0.8 x 10-lgGeV-l MP = 5.3 10-44s = 1.6 1 0 - 3 3 ~ ~ ~ =
-
M p M 101’GeV,
(1.4a) (1.4b)
where M p is called the Planck mass. It is clear from Eq. (4b), that the gravitational interaction becomes significant at Planck mass or
Fundamental Force
3
at a distance of order cm. Assuming that this interaction is of the same order as the electromagnetic interaction (see below) ( a = e l / & = 1/137, e l = e2/4mo) at Planck mass M p , we conclude that the effective gravitational interaction at 1 GeV is given by QGN
= (1GeV)2 M;
M
In particle physics, the gravitational interaction may be neglected at the present available energies.
ii. The Weak Nuclear Force It is responsible for radioactivity, e.g.
and
-, N~~+ e- + ve.
014
The latter process has half life of 71.4 sec. From the half life, we can determine its strength which is given by the Fermi constant [see Chap. 21:
GF x 10-5GeV-2 1
dG
x 300 GeV,
6x
0.7 x 10-16cm.
(1.6)
This is the energy scale at which the weak interaction becomes significant i.e. of the same order as the electromagnetic interaction. At an energy scale of 1 GeV,
aw =
(1 GeV)2 2cy = 10-sa. (300 GeV)
4
Introduction
iii. The Electromagnetic Force It acts between any two electrically charged particles, e.g. a negatively charged electron and a positively charged proton attract each other with a force which is proportional to their elec.t,ric charges. It is responsible for the binding of atoms and mainly governs all known phenomena of life on earth. This force also manifests itself through t,he electromagnetic radiation in the form of light, radiowaves and X-rays. Photon is a quantum of electromagnetic force and it, is the mediator of electromagnetic force, which is a long range force. The electromagnetic potential energy is given by
For electron and proton bound in hydrogen atom, this force of attraction provides the binding energy of the electron in the hydrogen atom given by Bohr’s formula
where p is the rediiccd mass of the system. For hydrogen atom p M me and mec2 = 0.5MeV, giving t,he binding energy of the electron lEll z 14 eV. For a proton ( p ) - antiproton ( p ) hypothetical atom ( p = m,/2 M 1000 me)so that, the binding energy provided by the electromagnetic potential is IETI M 14 keV. We see that the strength of electromagnetic interaction is determined by the dimensionless number cr = e k / h c = 1/137. In rationalized Gaussian units: e2 - e2 1 -a=4xhc 4x 137
iv. The Strong Nuclear Force It is rcsponsible for the binding of protons and neutrons in a nucleus. It is a strong force. We have scen that the electromagnetic binding energy for the p p atom is of the order of 14 keV, but, the
Fundamental Force
5
binding energy of deuteron (bound n p system) is about 2 MeV. Thus the strong nuclear force is about 100 times the electromagnetic force. It is a short range force effective over the nuclear dimension of the order of cm. Hence we conclude that the relative strengths of the four forces are in the order of (1.10) The experimental results on the scattering of electron on nuclei can be explained by invoking electromagnetic interaction only. In fact the scattering of y-rays on proton at low energy is given by the Thomson formula:
The neutrino participates in weak interactions only as reflected by the extreme smallness of the scattering cross-section of neutrino on proton, viz Deep -+ e+n, which is given by uw ‘v 10-43cm2.Comparing the above cross-sections with the one for nucleon-nucleon scattering, which is of the order u~ II 10-24cm2,we see that the electron and neutrino do not experience strong interaction. We now briefly and qualitatively discuss the ranges of the three basic forces. Due to quantum fluctuations, an electron can emit, a photon and reabsorb it, as depicted in Fig.1. Such a photon can exist only for a time (1.lla)
where A E = is the energy of the photon. Since the unobserved photon exists for a time < it can travel at, most,
3,
R = -.C W
(1.llb)
Introduction
6
Figure 1 Electromagnetic force mediated by a photon.
Now w can be arbitrarily small and therefore R can be arbitrarily large i.e. the distance over which a photon can transport electromagnetic force is arbitrarily large i.e. electromagnetic force has infinite range. This is expected from the Coulomb potential eL/T. If we assume that weak interaction is mediated by a vector boson W in analogy with electromagnetic interaction (Fig. 2), then since weak interaction is of short, range, W must, be massive. The
Figure 2
Weak interaction mediated by a vector boson W .
maximum distance to which the virtual W-boson is allowed t o travel by the uncertainity relation is
Fundamental Force
7
The W-boson has been found experimentally in 1983 with a mass mw M 80 GeV/c2 as predicted by Salam and Weinberg when they unified weak and electromagnetic interactions (see below). Eq. (12a) then gives the range of t,he weak interaction as
Rw
Figure 3
197 x MeV - cm 80 x lo3 MeV
M
2x
cm.
(1.12b)
Strong nuclear force mediated by a particle of mass
mh.
If the strong nuclear force is mediated by a particle of mass mh, as
shown in Fig. 3, then its range is given by (1.13)
Since nuclear force has a range of cm, mh M 100 MeV/c2. Yukawa in 1935, predicted the existence of pion by a similar argument. A particle of this mass was discovered in 1938, but it turned out that it was not the Yukawa particle, the pion; it did not interact strongly with matter and therefore is not responsible for strong nucleon force. It was actually the muon, while the pion was discovered in 1947 in the decay
Introduction
8
where v p is the neutrino corresponding to muon. The mass of mK was found to be 140 MeV/c2. Thus
R~ M 1.4 x
Jzf,
where f is called the Fermi and is equal to
1.2
(1.14)
1 0 - l ~M ~ ~
cm.
Classification of Matter: Leptons and Quarks
The electron and neutrino do not, experience strong interactions. They are just, two members of a family called leptons. The particles which also experience strong interactions are called hadrons. The proton and neutron are members of a much larger family of hadrons. There are six known leptons as given below: Leptons Ve,
e-
v/L, / A vT,
'?-
Mass [ 13
m,, < 15eV me M 0.51 MeV mvr < O.17MeV m, M 105.6 MeV mv7< 18.2MeV m, z 1777 MeV
Electric Charge 0, -1 0,-1
0, -1
Life Time
u, Stable re > 4.3 x loz3 yrs v p Stable T~ = 2.197 X S v, Stable rT = (290.0 f 1.2) x 10-15
Hadrons can be divided into two classes: (a) Baryons: They are fermions with half integer spin i.e. J = 1/2, 3/2. (b) Mesons: They are bosons with integral spin i.e. J = O,1,2. Pions T' , T O are the hadrons with lightest, mass (140 MeV) with J p = 0-. Hadrons found in nature are not, fundamental constituents of matter. There are hundreds of them. The experiments, for
Classification of Matter: Leptons and Quarks
Quark type (Flavor)
Electric charge
4 ( c ,4
(2/3, 4 / 3 1 (2/3, 4 / 3 1 (2/3,-1/3)
(U,
( t ,b)
Mass [effective mass or constituent mass in a hadron] 0.33 GeV (1.5 GeV, 0.5 GeV) (175 f 5 GeV, 4.5 GeV)
9
Introduction
10
- nucleus collisions, we can split, a niicleiis into its constituents viz. neutrons and protons. But, in high energy hadron - hadron collisions, a hadron is not, split, into its constituents viz., quarks. A hadron - hadrori collision results not into free quarks but into hadrons. This leads us to the hypothesis of quark confinement i.e. quarks are always confined in a hadron. The quark - quark force, which keeps the quarks confined in a hadron, is a fundamental strong force on the same level as electromagnetic and weak nuclear forces which are the other two fundamental forces in nature. Its strength is characterized by a dimensionless coupling constant a , = g,2/47r M 0.5 at present energies. It is actually energy dependent. The strong nuclear force between protons and neutrons should then be a complicated interaction derivable from this basic quark-quark force. As for example, the fundamental force for an atomic system is electromagnetic force, the interatomic and intermolecular forces are derivable from the basic electromagnetic force.
1.3 Strong Color Charges We have seen that, the quarks form hadrons; the baryons and mesons in the ground state are composites of (qqq)L,=o and ( q q ) L = o . Quarks and (anti-quarks) are spin 1/2 fermions. Now q and spins may be combined to form a total spin S, which is 0 or 1. Total spin for qqq system is 3/2 or 1/2. Further as q and have opposite intrinsic parities, the parit,y of the qij system is P = (- 1)(- 1)”= - 1 for the ground state. Thus we have for the ground states [see Chap. 61
11
Mesons
I
Bar yons
I
t] J p = 0-, 1- 1 J p = 1/2+,3/2+ 1 I
I
,
I
11
Strong Color Charges
Examples: Mesons
Baryons
~
-
There is a difficulty with the above pictaure; consider, for example the state, In++ ( S z = 3/2)) IuTuTuT). This state is symmetric in quark flavor and spin indices (T). The space part of the wave function is also symmetric ( L = 0). Thus, the above state being totally symmetric violates the Pauli principle for fermions. Therefore, another degree of freedom (called color) must be introduced to distinguish the otherwise identical quarks: each quark flavor carries three different strong color charges, red(r) , yellow(y) and blue@) i.e.
a = r, y,b
Q = Qa
[Leptons do not carry color and that is why they do not take part in strong interactions]. Including the color, we write e.g.
so that the wave function is now antisymmetric in color indices and satisfies the Pauli principle. Other examples, as far as quark content is concerned, are 1
12
Int roductiori
i.e. these states are color singlets. In fact, all known hadrons are color singlets. Thus, the color quantum number is hidden. This is the postulate of color confinement mentioned earlier and explains the non- existence of free quark ( q ) or such systems as (qq)l (qqq), and (qqqq). Actually nature has also assigned a more fiindamental role to color charges as we briefly discuss below. 1.4 Fundamental Role of “Charges” and the Standard Model of Electroweak Unification and Strong Force First, thing to note is that, the electromagnetic force and the strong nuclear force are each characterized by a dimensionless coupling constant and thus to achieve unification there has to be a “hidden” dimensionless coupling constant associated with the weak nuclear force which is related to the “observed” Fermi coupling constant, by a mass scale. That this is so will be clear shortly. Secondly we know that the electromagnetic force is a gauge force describeable in terms of electric charge and the current associated with it. This force is mediated by electromagnetic radiation field whose quanta are spin 1 photons, the mediators of electromagnetic force. This is generalized: All fundamental forces are gauge forces describeable in terms of “charges” and their currents as summarized in Table 1. Note that, the coupling constants a , a2, C Y ~in Table 1 are dimensionless but they are energy dependent due to quantum effects, a fact which is used in the unification of the forces. Note also that Q3 is not, identical with Qem;thus the unification of electromagnetic and weak nuclear forces needs another charge, call it Q B an associated mediator, call it, B , which does not change flavor like photon:
13
Fundamental Role of “Charges”
Then the photon y associated with the electric charge Qemis a linear combination of the mediators B and W . bosons, associated with the charges Q B and Q3 respectively,
y = sinOWW3 + COSOWB,
(1.15a)
while the second orthogonal combination Z = cos%wW3- sin%wB,
(1.15b)
is associated with a new charge Q z . 2 is the mediator of a new interaction, called neutral weak interaction. The weak mixing angle %W is a fundamentpalparameter of t,he theory and in terms of it,
Qz
= Q3 - sin2ow &em,
(1.16)
The weak color charges Q w , Qw and Q B generate the local group SUL(2) x U(1)where the subscript L on the weak isospin group SU(2) indicates that we deal with chiral fermions that is to say that the left handed fermions [i.e. those which appear to be spinning clockwise as viewed by an observer that they are approaching] are doublets under SUL(2) [required by parity violation in weak interactions] while the right, handed fermions (spinning anticlockwise) are singlets as indicated below:
(1.17)
We have
[ 1 3 ~is
the same as Q3 and $Ywis identical with QB] &em
=I
+1
~ L~ Y w ,
( 1.18a)
giving
-1_e2
1 3 + - 1
Q2
1
d2
(1.18b)
Introduction
14
L Force
Charges
Electromagnetic
Mediators of force: Spin 1 gauge particles
Coupling between basic fermions and mediators
Photon ( 7 )
a = -e27r
tWeak Nuclear
Qw,Qw [Qw, Qwl = Q3
# Qem
L Strong
3 color charges
w+,w-, WO W*change flavor as shown in the next, column
8 color carrying gluons: Gab a,b = 1,2,3
Fundamental Role of “Charges”
15
so that, we have the unification conditions e2
sin2Bw = T , 92
cos2ew
=
e2
-.
(1.18~)
912
Unlike photon, which is massless, the weak vector bosons W+,W and 2’ must be massive since we know that, weak interactions are of short range. This is achieved by spontaneous breaking of gauge symmetry (SSB). For this purpose it, is necessary to introduce a self-interacting complex scalar field
which is a doublet, under S u ~ ( 2 and ) has Yw = 1. This so-called Higgs field also interacts with the chiral fermions int,rodiiced earlier as well as with gauge vector bosons, W*, W3 and B . The scalar field 4 develops a non-zero vacuum expectation value:
thereby breaking the gauge symmetxy of the ground state 10). This amounts to rewriting
where $+ and hermitian fields $1 and 4 2 have zero vaciiiim expectation values. In contrast t,o the gauge invariant, vertices shown in Table 1 [which are not affected by SSB], one starts with manifest,ly gauge invariant vertices involving 4 and other fields and then translate them to physical amplitudes after SSB as pictorically shown on p. 16 [the dotted lines ending in X denotes (4) = 11/&]:
16
Introduction
Because of mixing between W3 and B, these are not physical particles, the physical particles y and 2 are defined in Eq. (15). This requires diagonalization of the mass matrix for W, - B sectors, which on diagonalization gives mA=0,
mZ=-,
mW
cos ow
(1.19a)
Fundamental Role of “Charges”
17
where from the above picture
1 mw = - g 2 u . 2
(1.19b)
Further we note from the above picture that the mass of a fermion of flavor f and that of Higgs particle H are respectively given by
hjq) mf
=Jz‘
mH =
m.
(1.19c)
What has happened is that $* and $2 have provided the longitudinal degrees of freedom to W* and 2 which have eaten them up while becoming massive. The remaining electrically neutral scalar field is called the Higgs field and its quantum is called the Higgs particle which we have denoted by H in the above picture. We note from Eq. (19a) that
The directly observed Fermi coupling constant in weak nuclear pro) by [cf. Eq. (18c)l cesses at low energies (i.e. << m ~is given
GF -922 - 8rnb
Jz
or mw= 2
[
e2
8m&sin2Bw’
(1.20b)
(1.204
&
where a = = is the fine striictiire constant, and GF is the Fermi constant (M lop5 GeVp2). The main predictions of the electroweak unification are (i) existence of a new type of neutral weak interaction mediated by Zo.
Introduction
18
(ii) weak vect,or bosons Mf, 2 whose masscs are predicted by the relations (20a) and ( ~ O C ) ,once sin2Bw is detmmined, a and Gr;.being known. (iii) existence of the Higgs particle with mass mH = is arbitrary since X is not, fixed.
m,which
The first prediction was verificd more than 16 years ago and the phcriorncnology of noiitral weak interaction gives sin2OW
M
(1.21)
0.23.
One can now iise t,his result to predict, m w and relations (20) to get
mNr= 80 GeV,
m Z M 92
GeV
m Z
through the
(1.22)
in agreement, wit,h their cxperinient,al valiies. The standard model is in very good shape experimentally. The third prediction is not, yet, tested and the present, lower boiind on mH from Higgs searches a t LEP is mall
> 77.5 GeV.
(1.23)
We also notc that, the elect,roweak unification energy scale is given by
M
250 GeV.
(1.24)
We will briefly discuss the unification of the ot,lier t,wo forces with the elcct,roweak force after discussing t,hc origin of the strong force between the ttwo quarks below.
Strong Quark-Quark Force
1.5
19
Strong Quark-Quark Force
We have already remarked:
(i) each quark flavor carries 3 colors. (ii) only color singlets (colorless states) exist as free particles.
Strong color charges are the sources of the strong force between two quarks just as the electric charge is the source of electromagnetic interaction between two electrically charged particles. To carry the analogy further, we note the following:
Electromagnetic Force Between 2 Electrically Charged Particles
Strong Color Force Between 2 Quarks
We deal with electrically neutral atoms. Mediator of the electromagnetic force is electrically neutral massless spin 1 photon, the quantum of the electromagnetic field. Exchange of photon gives the electric potential:
We deal with color singlet systems i.e. hadrons.
= Z-,. = lri - rjl 4a r For an electron and proton ve 23
Mediators are eight massless spin 1 color carrying gauge vector bosons, called gluons. Exchange of gluons gives the color electric potential:
I/?P = --4a 1 a - 2 ‘3 3 ‘ T 1 ’ - 4ff’ for qq color singlet system (mesons) while for qqq color singlet system (baryons). v.sq = -2 1 23
3&3T
20
Introduction
This attractive potential is responsible for the binding of atoms. The theory here is called quantum electrodynamics (&ED). Due to quantum (radiative) corrections, a increases with increasing momentum transfer Q 2 , for example 1 a(me)= 137, 1 4 m w ) = 128
(m)
Note the very important, fact that, in both cases, we get, an attractive potential. Without color, Vqq 23 would have been repulsive. The theory here is called quantum chromodynamics (QCD).
Due to quantum (radiative) corrections, a decreases with increasing Q2 [this is brought, about by the self interaction of gluons (cf. Table l)],for example a (m,) M 0.35, a , (my E 10 GeV) M 0.16, ( m ~M )0.125. That the effective coupling constant, decreases at, short, distances is called the asymptotic freedom property of QCD.
(m)
The binding energy provided by one gluon exchange potential of the form mentioned above cannot be siifficient, to confine the quarks in a hadron since a s one can ionize an atom to knock out an electron, similarly a quark coiild be separated from a hadron if sufficient energy is supplied. Thus Vg, the one gliion exchange potential, can at, best, provide binding for quarks at short distances and cannot, explain their confinement i.e. impossibility of separating a quark from a hadron. The hope here is that the self interaction of color carrying gliions may give rise to long distance behavior of the potential in QCD completely different, from that in QED, where the electrically neutral phot,on has no self interaction. One hopes that, the long range potential in QCD would increase with
Grand Unification
21
the distance so that the quarks would be confined in a hadron. Phenomenologically, a potential of form
Kj(r) = l$(r) + V c ( r ) , where
+
(1.25a)
(1.25b) . , r (... denotes spin dependent terms, (see Chap. 7 ) and k, = 4 / 3 ( q q ) , 2/3(qqq)) is the single gluon exchange potential while V c ( r )is the confining potential (independent of the quark flavor), has been used in hadron spectroscopy with quite good success. Lattice gauge theories suggest V c ( r )= kr, (1.25~) q ( T )
with Ic
=
QS
-ks-
* *
0.25 (GeV)2,obtained from the quarkonium spectroscopy. To sum up the most striking physical properties of QCD are asymptotic freedom and confinement, of quarks and gluons. The quark hypothesis, the electroweak theory and QCD form the basis for the "Standard Model" of elementary particles to which most of the book is devoted while Chap. 18 is concerned with the interface of cosmology with particle physics. We now briefly discuss the attempts to unify the other two forces with the electroweak force.
1.6
M
Grand Unification
The three strong color charges introduced earlier generate the gauge group SUc(3) while that of the electroweak interaction is s U ~ ( 2xU(1). ) Thus the standard model involves
- SUc(3) x a,
SUL(2) a2
x
U(1) a'
where the associated coupling constants a,, a2 and a' are very different at the present energies. But, these coupling constants are
22
Introductiori
energy dependent, due to quantum radiative corrections. Grand unification is an attempt to find a bigger group G:
G 2 SUc(3) x SU1,(2) x U(1) such that at somc energy scale q2 = m:, a,s(rn?J
=
a,(rnR) = Q / ( T r & )
-
QG.
(1.26)
This is possible because of the form of their energy dependence as calculated in qiiant,um theory (renormalization group analysis), see Appendix B and the fact that two of the three coupling constants, namely, a' and u2 are related at, fl = m w through the electroweak unification conditions given in Eq. (18c). As will be discussed in Chap. 17, the relation (26) holds at rnx x 1015 GeV or less, which gives the grand unification (GUT) scale. The most dramatic conseqiience of popular GUT models is that the proton is not, stable. How proton decay comes about can be seen as follows: in GUT, quarks and leptons share the same representation(s) and since gauge theories contain mediators linking all particles in a miiltiplet,, there are additional mediators (apart from the ones mentioned in Table 1 called lcpto~-quarksX ,Y (carrying 1/3 )) which tmrisforni qiiarks having strong charges f 4/3 and color charges to leptons as shown in Fig. 4. From dimensional analysis, lifc-time for proton decay is of the order [.-a is proton 1 GcVl. mass
m=
-
which gives 011 using a: z rp
-
and mx
M
1015 GeV (or less)
1031 years
(or less). This prediction is not yet borne out by experiment,. In fact the world's largest (IBM) detector sensitive to the decay mode years. In spite of this p -+ e+d' gives r ( p -+ e+.rro) > 5 x GUTS have some attract,ive fcatmes:
Grand Unification
23
1 1 qlmeson
r
-J 4
4
proton
R, Y
Figure 4
I
Decay of a proton via lepto-quarks.
quark-lepton unification (ii) relationships between quark and lepton masses (iii) quantization of electric charge, for a simple group it, is a consequence of the charge operator being a generator of the group and traceless. So for example, sum of charges in a multiplet containing quarks and leptons = 0, thus giving some relation between quark and lepton charges. (iv) may in principle explain the baryon excess of the universe, nB/n7M lo-'' and that there is no evidence for existence of antibaryons (see Chap. 18).
But they still leave arbitrariness in Higgs sector needed to give masses to lepto-quarks and W*, 2 vector bosons, do not, explain number of generations, do not explain fermions mass hierarchy typified by mt/mzlM lo5 and the gauge hierarchy problem mw/mx x lo-'' in a natural way. These mass hierarchies are more naturally accommodated in supersymmetry (see Chap. 17).
24
Introduction
1.7 Units and Notation We shall use the natural units: ti=c=l. We note that
[ti]
=
[c] = [tic] =
ML2T = 6.582 x 10-22MeV-s L T - ~= 3 x 10lOcm/s 197 x 10-13MeV-cm.
If ti = c = 1, then
If we take A4 = 1 GeV, 1 tic -- M 2 x 10-l~cm GeV 1000 MeV 1 ti T -- M 6.58 x GeV 1000 MeV 1 MeV = 1.6 x 10-6erg = 1.6 x l O - I 3 J 1 gm = 5.61 x 1023GeV 1 GeV = 103MeV,
L
N
N
we will denote the position by a 4-vector x ( p = 0,1,2,3) : xp
=
=(LA)
xp = (ct, -x) = ( t ,-x) = gpvxu 2 2 = xpxp = t2 - x2 with gpv cone
=
01
P
# v,goo
= 1,911 = g22 =
933
x 2 = o i.e. t 2 - x2 = 0.
=
- 1. On the light
Units and Notation
25
The energy E and momentum p are represented by a 4-vector p :
p2 = p,pc” = p ,2 - p
2
For a particle on the mass shell
E 2 = p2 + m2
The scalar product
= E2
- p2
Introduction
26
1.8
Bibliography
Bibliography for topics mentioned in Sections 1.5 - 1.6 which will either be not, developed or briefly discussed in t h e subsequent chapters:
1. A. Zee, The unity of forces in the universe, Vol. 1, World Scientific, Singapore (1982). 2. R. N. Mohapatra, Unification and supersymmetry, The frontiers of quark-lepton physics, Springer Verlag, Berlin (1986). 3. I. Hinchliffe, Ann. Rev. Nuclear and Particle Science, 36, 505 (1986). 4. M. B. Green, .J. H. Schwarz and E. Witten, Superstring theory I and 11, Cambridge University Press (1986). 5. L. Brink and M. Henneaux, Principles of String theory, Plenum (1987). 6. S. Dimopoulos, S. A. Raby and F. Wilczek, Unification of couplings, Physics Today, 44, 25 (1991). 7. R. E. Marshak, Conceptional foundations of modern particle physics, World Scientific, Singapore (1992). 8. M.E. Peskin, “Beyond standard model” in proceeding of 1996 European School of High Energy Physics CERN 97-03, Eds. N. Ellis and M. Neubert. 9. J. Ellis, “Beyond Standard Model for Hillwalker” CERN-TH/98329, hep-ph 9812235.
Chapter 2 SCATTERING AND PARTICLE INTERACTION Most of the information about the properties of particles and their interactions is extracted from the experiments involving scattering of particles. We, therefore, start this chapter by studying the kinematics of scattering processes.
2.1
Kinematics of a Scattering Process
Consider a typical 2-body scattering process
a+b+c+d. We denote the four momenta of particles a , b, c and d by pa, pb, pc, pd respectively. Energy momentum conservation gives: Pa
+ Pb = P c + P d
(2.la) (2.lb) (2.lc)
The reaction transition amplitude is a function of scalars (i.e. Lorentz invariants) formed out of the four vectors pa,pb,pc and pd. We assume Lorentz invariance in any process involving particles. The invariants are s =
t
( P a +Pb) 2 =
= (pa - P c ) =
2
( P a - pd)2
27
(pc +pd)2
(2.2a)
(pd -pb)2
(2.2b) (2.2c)
= (pc - Pb)'.
Scattering and Particle Interaction
28
Figure 1 Two-body scattering: a
+6
--f
c
+ d.
But only two of the three scalars are independent,:
In an actual scatt,ering experiment, we have a projectile (let it be u ) and a target ( b ) , which is stationary in the laboratory frame. Thus
Hence in the laboratory frame:
Kinematics of a Scattering Process
29
L
Figure 2
Two-body scattering in the laboratory frame.
or
(2.6~1) p i = - p g + u i = -ma+uL 2 2
(2.6b) (2.6~)
where X(2,y,z) = x 2
+ y2 + z 2 - 2 2 y - 22z - 232.
(2.7) Theoretically, it is convenient, to consider a scatkering process in the center of mass (c.m.) frame. In this frame:
Pa
=
(EarP),
Pb
( E c , ~ ,’ ) Pd
pc
= (Eb7-P) (Ed,-P’).
(2.8)
Thus we have s
=
(pa
+ P b ) 2 = (pc + pd)2 + Eb)2= (Ec -I-Ed)2
= (Ea
E&
(2.9a)
Scattering and Particle Interaction
30
Figure 3
Two-body scattering in the centre of mass frame.
Now
IPI =
d G 2&x i F 3
Similarly by considering, s = ( E ,
IP’l =
+ Ed)’, we get,
JG52-73 2 4
We also note that,
For elastic scattering
c-a,
(2.1la)
d z b
(2.1l b )
Interaction Picture
31
and
Ec
lpl = lp’l 7
E d = Eb
= Ea,
-4p 2 sin2 6-. (2.13) 2 Thus we see that -t is the square of momentum transfer. Finally we derive a relation between the scattering angles 0 and OL using Lorentz transformation. Let us take p~ and p along z-axis. The c.m. frame is moving relative to laboratory frame with a velocity: PL (2.14) V = VL mb Lorentz transformation gives
t
=
-2p 2 (1 -
=
+
p,“ cos OL = y b’ cos O p f s i n ~ L = p’sine
+ VE,] (2.15)
E,” = y [E, + VP’COSO]. Hence, we get p‘ sin 0 [p’cose VE,]
tanBL = where
y=
(2.16a)
+
1
- VL
+ mb
JC-7- Ec,
(2.16b) ‘
Equation (16b) follows from the relations: PL = [p
2.2
+VEa]
1
VL
= y[Ea
+ UP]
1
mb
= y [Eb - tlP].
(2.17)
Interaction Picture
In quantum mechanics, the transition rate from initial state li) to final state If) is given by
w = 2T I (fl v
12)
l2 Pf
(Ef) 1
(2.18)
where V is the interaction Hamiltonian viz.
H
= Ho
+ v.
(2.19)
Scattering and Particle Interaction
32
The above formula is obtained when V is treated as small in first order perturbation theory and li) and are eigenstates of Ho . p f ( E f )is the density of final states i.e. p f ( E f ) d E f= number of final states with energies between E f and Ef d E f . In order to define the t,ransit,ion rate in general, it, is convenient, to go to interaction pict,iire, which we define below: The Schrodinger equation is givcn by
If)
+
izd IQ ( t ) > s H IQ ( t ) ) s =
'
(2.20)
We now go over to interaction pict,ure by a unitary t,ransformation
IQ ( t ) ) ,= P
o t
IQ
( t ) ) ,.
(2.21)
Then, using Eq. (20), we have
Now define
An operator A^ in Schrodinger picture is related to operator & ( t ) in interaction pictiire by a unitary transformation
(t)
= ei H o t
e--iHot
(2.25a) (2.25b)
Scattering Matrix (S-Matrix)
2.3
33
Scattering Matrix (S-Matrix)
From the general principles of quantum mechanics, the probability of finding the system in state Ib) , when the system is in state ( t ) ) ,, is given by ICb(t)I2where
Cb@)= ( b IQ ( t > > . ~
(2.26)
,
Assume that I Q (t ) ) Iis generated from 1 Q (to)) by a linear operator
w, to):
I* (t>>l= U(t0, to)
U ( t , t o ) IQ ( t o h
= 1.
(2.27a) (2.27b)
Substituting Eq. (27a) in Eq. (24), we get (2.28a) so that we obtain 2
(t, to) dt
=
V I ( t ) U ( t ,t o ) .
(2.28b)
We note that U ( t , t o ) depends only on the structure of physical system and not on the particular choice of the initial state lq ( t o ) ) r . Thus
(2.29) Therefore,
U ( t , t’) U(t’, t o ) = U ( t , t o )
(2.30a)
I = U(t0,to) = U(t0, t ) U ( t , to) U ( t 0 , t ) = u-yt, t o ) *
(2.30b)
Thus, the operator
U satisfies the group properties.
(2.30~)
Scattering and Particle Interaction
34
T h e formal soliit,ion of differential equation (28b) is given
by
U(t,t,)
t
=
1-2 s VI(t’)U(t’,to)&’.
(2.31)
0
This integral equation can be solved by iteration. Thiis
Equation (32) is the basis of perturbation theory. Now at, t = t, ---t -w, the syst,em is known to be in an eigenstate) . 1 of Ho. Hence the probability amplitude for transition t o an eigenstate Jb) of Ho is given by
Now for t o -+
-cm, 19 ( t o ) ) s= la,to) =) . 1
(2.34)
e-iEato.
Hence from Eq. (33), we get
Our purpose is to calculate Cb(t)for large t (since for t system is an eigenstate of Ho) i.e. lim Cb(t)=lim (bl U ( t , -m)) . 1
t+cc
The operator
t+m
=
(bl U ( m , -cm) .)1
s = U ( m , -w)
-+
00,
tjhe
(2.36)
(2.37)
35
Scattering Matrix (S-Matrix)
with matrix elements
is called the S-matrix.
An important property of S-matrix is that, it is a unitary operator. This follows from the conservation of probability. Now (2.39) or
or
cI.(
S+)b) (bJs}.I
= 1.
(2.40)
b
Hence
I.( StS).1
=1
i.e.
S+S= i.
(2.41)
Therefore, S is a unitary operator. We can express the operator U ( t , t o )explicitly in terms of the Hamiltonian. A formal solution of the Schrodinger equation (2.20) can be written as
( t ) S) -- e - i H ( t - t o )
I*
(to))s.
(2.42)
Then using Eqs. (21), (27) and (42), we have
e -iHot u(t,t 0 ) e i H 0 t
= e-iH(t-to)
(2.43a)
Scattering and Particle Interaction
36
and this limit is taken with the following prescription:
U ( t , -00) =lim
E
e&tfeiHote-ifi(t-t')
e -2Hot'dtI .
(2.4410)
Similarly,
(2.44~) Hence we have
U ( 0 , -m)) . 1
=
[!%&S-,
0
e i H t ' e - i E, 1'
ie =
lim &-+Q E, - H
+ i~ I4 .
It is clear from Eq. (45), t,hat, in the limit
') . 1
= U ( 0 , -m) ).I
=
E -+
(2.45)
0, the state
).I + Ea - H1 -+ ZE V I4
(2.46)
is an eigenstate of H with eigenvalue E,. The state Iu') is called the incoming state or simply "in" state. The notation emphasises that the state lu+) goes over to the unperturbed state) . 1 as t -+ -00. Similarly, we define an "out" state
I.-)
= U ( 0 , 4 I4 -2E
= Elim - 4 E, - H - i€
I4 (2.47)
From Eq. (38),we get
=
(blu')+
1 Eb - Ea
+ 2& ( b J V aJ').
(2.48)
Scattering Matrix (S-Matrix)
37
Now
(2.49) where we have used Eq. (46).Hence we have from Eqs. (48)and (49) (2.50) Sba = fiba (Ea- Eb) ( b IvI , where we have used the fact
.'>
7rrS
(E, - Eb) =lim '--*O
&
(Eb - Ea)2
(2.51) E2
We define an operator T , called the T-matrix (transition matrix) with the matrix elements. Tba
= (b(T) . 1
= - ( b (1/(a ' ) ,
(2.52)
so that
S h = 6ba
27riS (Eb - Ea) Tba'
(2.53)
We note that (bl T).I
= - ( b IVI = - (bl V) . 1
.+)
>
1
- ( b ' V E a - H + i e V I u ) (2.54)
or
T=-V-V
1
Ea - H + i~
V.
(2.55)
In relativistic theory, where we treat energy and momentum on equal footing, we write for Eq. (53):
Sfi = Sfi
+ ( 2 ~ )2 6
4 . 4
(PP- pi) Tji,
(2.56)
38
Scattering arid Particle Interaction
where we have put) . 1 skates and
=
li), Jb) =
If)
to signify initial and final
4
(5 (Pf- Pi) = s3(Pf - PI) (5 (El. - E,) .
(2.57)
The 6-fiinction ensures the energy momentum conservation in the transition. Then, using Eqs. (36) and (38), thc transition probability for large t from a state li) to state for i # f is given by ~ = / i m[cf(t)l2 = = C(27r)'ti4(pf - p i ) s 4 ( 0 ) l ~2 .
If)
~(flsli)l~
-00
(2.58)
Now
(2.59) Therefore, the transition rate per unit macroscopic volume is given by P (2.60) Wf. - - = (27d4 s4 (Pf- Pi) ITfiI2 . -
c
Vt
To carry out, slim over final states, we need to know the density of final states pf ( E f ) . 2.4
Phase Space
Consider first a singlc particle in one dimension confined in the region 0 5 IC 5 L . The normalized eigenst,at,e of moment,iim operator fj is givon by
Thc boundary condition that, up(x)is periodic in the range L gives p =
(F)
n,.
(2.62)
~ ~
39
Phase Space
Thus
dn
(2.63)
i.e. the number of states within the interval E and E by d n = p ( E ) d E . In three dimensions, we have
(&)"$
p(E)= dn = dE
/d3p = (&)3p
2
+ d E is given
d~ dP / d o .
(2.64)
We now generalize to n particles in final state:
since pa = pf = p i
+ p ; + + p:, a
(2.66)
' '
and only (n- 1)momenta are independent. With the normalization L = 27r, we can write from Eq. (65)
n=
/ 63
[pi - ( p i
+ p ; + - . + p;)] d3p', d3p',
. d 3p,., +
(2.67)
Thus we can write
+ + + p:,)]
[Pi - ( p i p ; x d3p', d3p; * * * d3p',. Xb3
'
* *
(2.68)
Hence the transition rate [cf. Eq. (SO)]
Wf
=
1
( 2 ~ )d3p', ~ d3p', d3pL X ITfiI2b4 ( p i p2
C
+ + + p; - p i ) , (2.69) * * *
final spins
where - denotes the average over initial spins if initial particles are unpolarized, otherwise we have to use the density matrix if initial particles are polarized.
Scattering and Particle Interaction
40
Remarks: (1) In the first order perturbation theory
Tfi = - ( f l V
12).
(2.70)
(2) Our normalization of states is (P’lP) = p 3 x (P’IX) (XlP)
The phase space J d 3 p is not, Lorentz invariant. Thus we consider the Lorentz invariant, phase space
==
/g
(2.72)
Now we write
(2.73) It is clear from Ey. (73) , that (p’JT lp) is not, Lorentz invariant 0 po (p’( T lp) is. Thus in general we write but
Tfi = N‘Ffi,
,
(2.74)
where N’ is a miiltiple of fact,ors like l / E T .In fact, it, is convenient, to take
N‘=
(2.75)
41
Examples
if there are
T-
fermions and s bosons such that
m and n being the number of initial and final particles respectively. Hence finally the transition rate is
(2.77) In the first order perturbation theory
(2.78) For example for s = 0, r = 4
(2.79) Here ma, mb, m, and md are the masses of four particles a , b, c and d involved in a scattering or a decay process. 2.5
Examples
2.5.1 Two-body Scattering Consider the scattering process a+b+c+d, where a and c are bosons e.g. pions and b and d are fermions e.g. nucleons. The scattering cross section is given by da =
dW
(
’
(2.80)
Scattering and Particle Interaction
42
where (Fliix)i, is the incident, flux defined as (2.81)
(2.82) We calculate the scattering cross section in the c. m. frame. In this frame: = Pab = p , =Ec+
Pa
=
-Pb
Ecm
=
E a + E b
p c
=
-Pd
(2.83a) (2.8313)
= P c d = PI
Ed.
(2.84) Now from Eq. (77)
(2.85) spin
We can write 4
=
6 (pc+Pd-Pa-pb) 3 6 ( P c P d - P a - P b ) 6(Ec -I- Ed - E a - E b )
a
(2.86)
The integration over d 3 p d in Eq. ( 8 5 ) can be removed by t,he threedimensional &function. Writing (2.87)
d3p, = Ip1I2d (p’ldR’,
we have from Eq. (85)
/
\
43
Examples
Now using the formula (2.89) we have from Eqs. (84) and (88)
do =
dW ( 2 ~ ) ~ Vin
(2.90) where we have put (2.91) spins
Hence we have
do mbmd lp’l /MI2 -=---
EL
dQ’ If a and c are also fermions, then
u -d=
Ip( 1 6 ~ ~ ’
mambmcmd
dR‘ Jqk 2.5.2 Three-body decay Three-body phase space.
-lp’l [MI2 (PI 4T2
Consider a three-body decay m
ml+m2+m3
K = Pl+P2+P3. The decay rate [cf. Eq. (77)] is given by [pin = 4 (2x1 1 dr =
dW
Pin
(2.92)
(2.93)
Scattering and Particle Interaction
44
where for definiteness, we have taken all the particles to be fermions. We evaluate Eq. (94) in the rest frame of particle m. In this frame K = 0 and E = m. Hence we have P1
+P2+P3
El + & + I 3 3
= 0 = m.
(2.95)
From Eq. (94), removing the integration over d3p3 due to threedimensional &function, we get
After performing the angillar integration over
Rl2
, we obtain
(2.97) -2
where IM1 is the value [MI2after the angular integration has been performed. In order to evaluate the integral in Eq. (97), it is convenient to define the invariants:
In the rest frame of particle rn, we have s12
= m2+m:-2mE3
m2+mi-2mE2 = m2+m:-2mE1 = m2 rn? mi mi.
~ 1 3=
523
s12
+ s13 +
s23
+ + +
(2.99) (2.100)
Examples
45
On the other hand, in the center of mass frame of particles 1 and 2, we put p1 = -p2 = p and p3 = q. (2.101) In this frame, we denote the energies of particles 1, 2 and 3, by w1, w2, w3 respectively. Thus in this frame 2
+ q ) = rn? + m: - 2p.q+2w1w3 - (p - q> = mi + mi + 2p.q+2w2~3
s13
= (w1+ w3) - (p
s23
= (w2
s12
=
+ w3)'
2
2
(2.102)
(Wl + ( J 2 y .
For fixed s I 2 , the range of ~ 2 3is determined by letting q to be parallel or antiparallel to p. Thus
We also note that we can express wl,w2 and w3 in terms of
s12.
(2.104) In terms of the invariants
513
and
~ 2 3 Eq. ,
(97) can be written (2.105)
The scatter plot in
523
and s12is called a Dalitz plot (Fig.
1- l2
4). Phase space density is uniform across the plot i.e. if M
is a constant, we have uniform distribution of events. Non uni orm distribution of events over Dalitz plot will indicate a structure in and would provide an important information about the dynamics underlying the process concerned.
]&?Iz
Scattering and Particle Interaction
46
Figure 4
Dalitz plot for a three-body final state [ref.
51.
47
Examples
p- decay.
e.g. 014
+ N14
+ e- + D,,
We obtain from Eq. (94)
(2.106)
Now Pu
dPu = Eu dEu.
(2.107)
It is a very good approximation to neglect the recoil of the particle B, so that p B M 0 and EB = mE. For this case, the &function removes the integration over dEu and we get
(2.108)
Let us write
Emax= E,
+ E,
M
mA - m g .
(2.109)
Then we get
x
( Ee Eu
me mu
(MI'> d o e dR,.
(2.110)
Scattering and Particle Interaction
48
In the first, order pert,lirbat,ion theory,
\MI2= (27dL2C I ( f l Hw li>l2 -.Ee spin
1(fl
If the expression
7%
E u
(2.111)
m u
H w li) l2 is averaged over angles between
spin
electron and neiit,rino, d r can be integrated over dReu and we obtain
spin
(2.112) We make the simplest assiimption that, the averaged expression is independent, of electron energy Ee . In this case
(2.113)
If we neglect, the mass of the neutrino, then (2.114) From Eq. (114))wc see that, plot, of ( d r / p : dpe)1’2vcrsus E, should be a straight line. This is called Ferrni or Kurie plot,. Figure 5 shows that, it is indeed a straight, line. Therefore, our assumption that the matrix elements ( f l Hw li) are independent, of energy is correct. From Eq. (114)) we get,
ra
1 7 ( 2 4 [(2x)’2 l(fl ~w li)12] 0 1 m5 (W12 (fl Hw li)12] (27rI3
=
[
I
( ~ r n a x-
f
(Po) 7
2
2
Ee) p ,
diDe
(2.115)
Examples
49
K
Figure 5
Fermi or Kurie plot.
where
This does not take into account Coulomb corrections due to Coulomb force which the electron experiences with the nucleus of charge Ze once it has left the nucleus. This can be taken into account in the integral f ( P O ) and the formula (115) remains valid. The life time for @-decay rp = l / F p , but it is the half life t l p = ~ p ( l n 2 which ) is experimentally measured, while f is computed. f t l l 2 is called the ft value. It is assumed that Hw is universal i.e. the same for all decays (this assumption is supported
Scattering and Particle Interaction
50
Table 2.1 Some characteristic f t values.
I I Decay
1/2 --t 1/2 He6 -+ La6 0 4 1 0 1 4 -+ "4 0-0 1/2 --+ 1/2
T,maz
ill2
= Emax
f tll2 - me
(Set>
(MeV) 10.6 min. 0.782 0.813 sec. 3.50 71.4 sec. 1.812 12.33 vr. 18.6 x 10-3
1100 810 3100
by the experiments). f t values vary from about, lo3 to seconds. This variation is due to the phase space available in the final state characterized by Emaxand hence by f ( P O ) . Other cause of variation is due to the nuclear wave functions that enter into the calculation of matrix elements (fl Hw 12). Without the universality of H w , an understanding of weak interaction would be hopeless. Some characteristic ft values are shown in Table 1. We now consider the transition 014-+ N14 (0 -+ 0)so that we do not have complicat,ionsdue to spin. Nuclei may be described by highly localized wave functions described by *Ui(r) and Pa)
*Uf(r) which vanish for r > cm. Electron and neutrino can be described by plane waves as they carry large momenta. We take that, Hw responsible for ,&transitions is characterized by a parameter GF which determines its strength. Thus
~ ( f l Hw li)12 = Gg
/T.;i" / 1
u;(T)
ui(r)ezp6.r
e i p v . r d 3 r12.
(2.117) Since pe/h 10l1 cm-', T M cm, it is a good approximation to replace the exponential in the integration by 1. This is called the allowed approximation. Thus we get from Eq. (117) N
(2.1 18) Hence we obtain
G; m,5 rP -- 2,rr3 fko)
(2.119)
Examples
51
(2.120) or
(2.121) Using
$ = (0.7)
sec. and f t = 3100 sec. we get
GF m i
M
1.5 x
(2.122)
=
(2.124)
More careful calculation gives
so that
GF m$
Finally we note from Eq. (113) that a non-vanishing neutrino mass reveals itself as a downward deviation from a straight Kurie plot as the energy approaches its nominal (m, = 0) kinematically allowed maximum T,mal. We can write Eq. (1 13):
where
T, = E, -me = d m - m e .
(2.126)
We note that the effect of m, is near Te = T,""", otherwise (T,""" Te)2>> m;. If we put Te --
-2)
TFX
z e = - me T F X'
(2.127)
then
/ x3/2
rv=ooc ( T y ) 5
1
0
(1 - x)
.(
+ 22,)3/2 dx. (X
+Xe)
(2.128)
Scattering and Particle Interaction
52
Hence it follows from Eqs. (125) and (128), that, if m, # 0, then the fraction of events G (m,) which will be absent at the end point is given by
3
0:
( T y ) 2m:
=g
(Ty)5
(3) , (2.129) T,maz
where g is some constant. Hence it follows from Eq. (129), that in order to have G(rn,) as large as possible, T y be as small as possible. Thus we see from Table 1, that tritium ( H 3 ) is most suitable to determine the mass m, of neutrino experimentally, since electrons from this decay have very low end-point energy (18.6 keV). The distortion at the extreme end of the Kurie plot due to m, # 0 is shown in Fig. 6. Thus in order to determine m, one has to look for such a distortion, but note that the deviation is in fact quite small. Moreover, the fraction of the events in the energy range of 18.5 keV 5 Ee 5 18.6 keV is only 3 x The experiment is hence quite difficult and even then it would be extremely difficult to determine rn, better than 10 eV by this method. We shall come back to this point in Chap. 9. 2.6
Electromagnetic Interaction
A mono-chromatic electromagnetic wave is composed of N monoenergetic photons, each having energy and momentum, E = tiw, p = hk. The electromagnetic field is described by a vector potential A with polarization vector E . Electromagnetic waves are transverse waves so that k . E = 0 and these waves have two independent states of polarization. We can conveniently describe it as left-circularly or right-circularly polarized photon or we can say that a photon has two helicity states f l . Such a photon can be
Electromagnetic Interaction
53
\ OeV I
18.4
*
*
I
I
~
~
'
"
'
'
'
'
"
'
18.5
18.6
KINETIC ENERGY (keV) Figure 6 A schematic drawing of the Kurie plot with neutrino mass of 0, 10 eV and 30 eV.
Scattering and Particle Interaction
54
described by a polarization vector €* =
1 - (Tl , -i, O ) ,
E
Jz
0
= 0,
(2.130)
where we have taken the propagation vector k along z-axis. The spin 1 matrices S are given by (si)jk
(2.131)
= i&ijk’
Writing them explicitly, we have
s,
=
s,=
(
0 0
-i
0 2 0 0 ) 0 0
(2.132)
If we write
E+
and e- as column matrices
it is easy to see that they are eigenstates of S, with eigenvalues f l respectively. We also note that
1= r t
e;.e+
=
€;me-
= o=e:
€1 € A *
= 6AXt
*€‘€+
, X,X’
=
& l a
(2.134) (2.135)
For a real photon, if we sum over polarizations (spin), we have
c X=fl
&IX&jX
= saj
ki kj k2
- -*
(2.136)
Electromagnetic Interaction
Figure 7
55
Electron-Electron scattering through exchange of a photon.
In quantum field theory, electromagnetic force between two electrons (or any charged particles) is assumed to be mediated by photons, the quanta of electromagnetic field. The simplest case is the exchange of a single photon as shown in Fig. 7. The Coulomb potential between two charged particles is e 2 / 4 m in rationalized Gaussian units. This is the Fourier transform of an amplitude M ( q ) corresponding to the diagram shown in Fig. 7. Thus we write
(2.137) In order to find M(q), we note that
Hence we have (2.139)
Scattering and Particle Interaction
56
This gives the matrix elements of the above diagram (Fig. 7) in momentam space in non-relativistic limit,. A relativistic generalization of this is
i.e.
(2.141) where
(P) is the expectation value of the electromagnetic current,.
g F y / q 2 is called the Feynman propagator of the photon. J p is given
by
Jp=e
yp
@,
so that, in free particle approximation
( J ” ) i = ea (p:) 7% (pi) ,
2
= 1,2.
(2.142)
Thus
(2.143)
In the non-relativistic limit
Ei
M
m,
<
a (P>You. (P) = 1, q2 = ( p i
-
5 2
M
0,
M
0 , El
=
E2 = Ei =
= 01 p l ) 2 = (Ei - El) 2 - 4p2 = -4p 2 , ‘ci (P) YU (P>
and q2 --+ -q2 so that we have from Eq. (140)
(2.144)
57
Weak Interaction
2.7 Weak Interaction
If weak nuclear force is mediated by exchange of some particle, then this particle must have a finite mass, since weak nuclear force is a short range force. We assume that mediator of this force is a vector particle of finite mass. It, therefore, has three directions of polarization or it is a spin 1 particle with M, = f 1 , O . These spin states can be expressed as Q
1
-i, 0 ) ,
= -(TI,
Jz
&$
=
o (2.145)
In this representation 4 = (40,0, 0,
Isl) , q2 = m2,
(2.146)
so that
(2.147)
q ' & = 0.
It is easy to see that E & , €0 are eigenstates of S, with eigenvalues f l , 0 respectively. For a spin 1 particle on the mass-shell
c &;& &y&; I + =
X=fl,O
&!&?*
+
& *;
=
-gPU
+"l q'.
(2.148)
m2w
In order to estimate the strength of weak interaction, we evaluate the matrix elements of the scattering process
given by the diagram in Fig. 8. In analogy with Eq. (141), the scattering amplitude F is given by [the propagator 1/ (q2) is replaced by 1/ (q2 - rnk) as W-boson is massive] ds
(2.149a)
Scattering and Particle Interaction
58
Figure 8 boson.
Neutrino-electron scattering through exchange of vector
Now in contrast to electron, neutrino is a two-component object and its wave function is (1 - 7 5 ) u ( p ) . Thus in analogy with Eq. (142) ( J w P ) i = gw Ti(pi) 7’ (1 - 7 5 ) pi),
i
=
1, 2,
(2.14913)
where gw is the strength of weak interaction just as e is the strength of the electromagnetic interaction. Thus for q2 << mk
(2.150)
Weak Interaction
59
(2.151)
where
+
s = (Pl p2)2 = (pi
+
= E&.
(2.152)
From Eqs. (93) and (150), we get (2.153)
If we neglect the lepton masses (viz. for s >> m:), then we have Q =
NOW
(C)'(&)
4x9.
(2.154)
GFI,& = g&/mb so that we have (2.155)
Taking
x
cm2 at s = (1 GeV)2,we get
GF
M
GeV2
(2.156)
to be compared with Eq. (124). This shows the universality of the weak interaction since GF is the same as obtained from the &decay or from the scattering of neutrinos on leptons. In unified electroweak theory [see Chap. 141 e
gw sinew = -
2 4 '
(2.157)
Scattering and Particle Interaction
60
Figure 9
Nucleon-Nucleon scattering through pion exchange.
where sin OW is a parameter of the theory. Experimentally, sin2 Ow M 1/4. Thus we get G F--
fi
e2 a = 47r 8m&sin20w 8m&sin2Ow
'
(2.158)
.
(2.159)
or 112
(sin20w G F ) - ' ] Using sin2 Ow
M
1/4, and Eq. (155), we get, mw
2.8
M
80 GeV.
(2.160)
Hadronic Cross-section
Consider the N - N scattering through the pion exchange. In particular consider the diagram (Fig. 9). Neglecting the spin of the nucleon (2.161) From Eqs. (93) and (160), we get du
- = 9, dR (q2
1 2mpJ---. 4 1 lP'l 1 - m:) 47r2 IpI Ec",
(2.162)
61
Problems
For elastic scattering lp’J= lpl, so that, 2n sin 8d8
[1+ W(1 m4, - COSQ)]
2
(2.163) Now
s = 4mk (1
+
2) ,
(2.164)
where Eff is the incident kinetic energy of the nucleon. Now 2x
cm2,
&
M
(2)’ 50, thus M
For Eff << % M 10 MeV, (2.166) Experimentally o x 5 x 10-23cm2, therefore, 2
-s -s 1 - 2 . 41r 2.9
(2.167)
Problems
1. Show that for the scattering e-e++y--+.rr
+
( kl)
r- (k2)
the differential and total cross sections are given by (s >> m;):
do -dR
(2 Q2 IF(s)12 S
8T
a =
-Q2p(S)]2
3s
312
- rn;)
(1 - cos20 )
s312
312
( 2 - mq)
s312
Scattering and Particle Interaction
62
+
where s = q2 = (kl k2)2 and F ( s ) is the electromagnetic form factor of the pion, defined by (01Ji”17r+(k1)7r-(k2)) = F ( s ) ( k ,+ k2)F
Hint: See Appendix A. 2. Consider the decay
Discuss the Dalitz plot for this decay. Hint: From Lorentz invariance, the decay amplitude
where pl,p2 and p3 are four momenta of pions.
Bibliography
63
2.10 Bibliography 1. G. Kallen, Elementary Particle Physics, Addison - Wesley, Reading, Massachusetts (1964). 2. S. Gasirowicz, Elementary Particle Physics, Wiley, New York (1966). 3. H. M. Pilkuhn, Relativistic Particle Physics, Springer - Verlag, New York (1979). 4. D. H. Perkins, Introduction to High Energy Physics (3rd Edition), Addison - Wesley, Reading, Massachusetts (1987). 5. Particle Data Group, The European Phys. Journal C 3 , (1998).
Chapter 3
SPACE-TIME SYMMETRIES 3.1
Invariance Principle
If the result of any experiment on some system is unchanged by a physical transformation of the apparatus, then the Hamiltonian or S-matrix describing that system is said to be invariant with respect to that transformation. In quantum mechanics, such a transformation is described by a unitary transformation. Consider a particular experiment, for example, a transition from an initial state 12) to a final state If). Such a transition is described by the matrix elements S 12). If we change (transform) apparatus (e.g. displace or rotate), then invariance means
(fl
(f or
1sI 2 ) = ( f"
ISI
2"
)
=
( f I U+SUI 2 )
(34
s = UtSU
(3.2a)
[S, U]= 0.
(3.2b)
or Here
are the transformed states. We see that the invariance under unitary transformation means that S-matrix commutes with it. Since
65
Space-Time Symmetries
66
S-matrix is related to the Harniltonian of the system, it follows that
[ H , V ]= 0
(3.4)
if the system is invariant, under U . We consider two cases:
(a) U continuous:
U can be built out of infinitesimal transformations. Thus we need to consider an infinitesimal transformation: A
U = 1- i e F ,
(3.5) where is a hermitian operator. can often be identified with an observable of the system, for example, the energy-momentum Pp or the angular momentum J . ? is called the generator of the transformation represented by U. From Eq. (2b), we get
[s,F] = 0. This means that p is conserved. To see this, let, li) and eigenstates of F :
F li) Flf)
=
F, li)
=
Ff If).
(3.6)
If)
be
(3.7)
From Eq. (6), we have
(fl [s,q 12) = 0
(3.8a)
or
(Fi- F f ) (fl S li) = 0.
(3.8b)
Hence, we get
Fi = Ff if (fl
s li) # 0,
(3.9) i.e. is conserved (eigenvalue of P is conserved) in the transition is then said to be a constant of motion. In Table li) to If). 1, we give a list of some of the common transformations and their generators. Invariance under these transformations means that the corresponding generators are conserved.
67
Parity
Table 3.1
Transformation
Generator of the Unitary transformation operator
Conservation Law
A
Displacement in [3] Momentum operator p x+x+a Displacement in [4] EnergyMomentum 21,+ x, a, operator Pp Angular Rotation in [3] Momentum J xa t xi = xa 4-eijxj 1 wi = 7j&ijk &jk
eip.a
-iP, a,
+
e-i~.J
Momentum is conserved EnergyMomentum is conserved Angular Momentum is conserved
(b) U is discrete (e.g. space reflection) Eigenvalues of U are
u2= 1.
(3.10a)
u' = f l .
(3.lob)
Thus U is both unitary and hermitian. U can be regarded as an observable.
3.2
Parity
Consider a transformation corresponding to space reflection:
x + xI = -x. The corresponding unitary operator is denoted by on a wave function gives
P Q ( x ,t ) = Q ( - x , t ) .
(3.11)
p, which acting (3.12)
68
Space-Time Symmetries
Now
P-2 = 1 , so that,
(3.13)
has two eigenvalues f l . If
[s,PI
=0
[HI P] = O
or
(3.14)
F
does not commute with all then we say that, parity is conserved. types of H . In particular, the weak interaction Hamiltonian Hw does not, commiite with :
F
(3.15) i.e. parity is not, conserved in weak processes. Under parity operator P
x---x,
p+-p
(3.16)
but, the orbital angular momentxm + L,
(3.17a)
u-0.
(3.17b)
L=xxp so t,hat.
J+J,
Such vectors are called axial vectors. Also under parity, the scalars:
(P1 X P 2 )
x.p--x.p
(3.18a)
P3+ - (P1
(3.18b)
'
X P 2 ) . P3
J.p+-J.p.
(3.18~)
The scalars which change sign under parity are called pseiidoscalars. All the three quantities are rotational invariant,, but the last two have different, behavior under I;. A particle when it is in an orbital angular momentum state 1 has an orbital parity associated wit,h it,. In polar co-ordinates x f ( r ,8,q5)) so that x + -x implies
r-r,
Qjn-8,
+-n++.
(3.19)
Intrinsic Parity
69
Now we can write the wave function of a particle as
(3.20b) Under space inversion y(C0se)
t
eirn&
+
ern(-
COSO) eim(&+*)-
K m ( 4 $1
--f
m im&
,
(3.21b)
(4K m ( 6 4).
(3.2lc)
-
so that
= ( - i ) ~ + r n ~ m ( c o s (3.21a) e)
(-1) e
We see that the orbital parity of a particle in an angular momentum state Z is (-1)'. 3.3
Intrinsic Parity
As far as orbital parity is concerned, it is independent of the species of particles and depends only on orbital angular momentum state of system of particles. When creation or annihilation of particles takes place, we have to assign an intrinsic parity to each particle. Consider, for example, a photon, the quantum of electromagnetic field represented by a vector potential.
A (4 = &
f (4>
(3.22)
where e is the polarization vector and f(x) is a scalar function. Now the interaction of a charged particle with electromagnetic field is introduced by the gauge invariant substitution: p --+ p-e A (x).
Since x and p change sign under
(3.23)
@,it follows that
A (x)---t -A (-x)
(3.24a)
70
i.e.
Space-Time Symmetries
...
A
P A (x) P-' = -A (-x).
(3.24b)
This means that under parity E --+
(3.25)
-&.
The behavior of the polarization veckor E characterizes what, we call the intrinsic parity of a phot,on. Thus we say that, intrinsic parity of a photon is odd. Similarly for any particle a represented by a state vector la,p) ,
1%
P) = 7:
1%
-P)
7
(3.26)
where 7: is called the intrinsic parity of particle a. Note that, 7: = fl . We now show that the conservation of parity leads to multiplicative conservation law. Consider a reaction
a+ b + c + d .
(3.27)
We can write the initial state
1 i ) = 1 a ) I b ) I relative motion ) .
(3.28)
Here I a ) and I b ) describe the internal states of a and b, while the third factor describes their relative motion. This state can be described by a wave function R(r)Y;, (0, 4). Since, we assume that parity is conserved in the reaction (27), it follows that the states I i ) and I f ) are eigenstates of p , with eigenvalues f a n d qfp respectively. Now
where q f , qbp, q:, and $are intrinsic parities of a, b, c and d respectively and (-1)' and (-1)'' are their orbital parities in the initial and final states.
Intrinsic Parity
71
Parity conservation for the reaction (27) gives P P rli = r l f
(3.30a)
or P rla
P
P
P
76 (-l)' = ??c q d (-l)''
(3.30b)
i.e. parity is conserved as a multiplicative quantum number. However, the law of parity conservation is not universal, in particular it does not hold for weak interactions. Then it follows from Eq. (15) that it is not possible to find simultaneous eigenstates of Hw and p . Thus if parity is not conserved, the energy eigenstates I Q ) are not expected to be eigenstates of parity. In this case, we can write (3.31) where IQregular) and IQirregular) have opposite parities. Y is called the parity mixing amplitude and is a measure of the degree of parity non-conservation. Parity violation is maximum if (yI2= 1. Several experiments involving hadrons show that, in hadronic interactions
Experiments involving atomic transitions show that parity is conserved to a high degree in electromagnetic interaction and that y l2 < For weak interactions, the parity violation is maximum viz. IyI2 = 1. It follows that in order to determine the intrinsic parity of a particle, one cannot use weak interact,ions. Only by considering reactions involving hadronic or electromagnetic interactions, one can determine the intrinsic parity of a particle. Even then the intrinsic parity cannot be fixed uniquely and we have to use a convention viz. the intrinsic parity of a proton is +1 i.e.
I
q (proton) = +1.
(3.32)
Since proton and neutron form an isospin doublet, we also take
q (neutron) = +1.
(3.33)
72
Space-Time Symmetries
Intrinsic Parity of Pion We shall assume that, the spin of pion is zero (we shall show later, how it, comes out to be zero). Consider first the decay 7~' -+ 27. Here we have two polarization vectors el and e2 corresponding to two y - rays, whose momenta we t,ake as kl and kz , such that (gauge invariance) kl.el = 0, k2.e2 = 0. We also note that ~ 3 1 . ~=2 0. Now only the momentum k = kl - k2 is independent as K = kl +kz = 0 in the rest frame of TO. It is clear that, the only invariant, which we can form is k . (el x e2), which is a pseudoscalar, showing that intrinsic parity of no is -1. Consider the captiire of n- a t rest, by deuteron. The dominant processes are n-+d
-+ -+
n,+n nr+n,+y.
(3.34)
Parity conservation for the first reaction gives
(3.35) where 1 is the relative orbital angular momentum of n-d and 1' is that, of two neutrons. There is evidence that, n- is captured in 1 = 0 orbital state. Thus from Eq. ( 3 5 ) , we get
(3.36) The deuteron is a bound state of a proton and neutron and has spin 1. The relative angular momentum of the two nucleons in deuteron is predominantly zero. Thus deuteron is a predominantly 3S1state i.e. for a deuteron J p = 1+. It follows that the total angular momentum of the initial state is J = 1. Conservation of angular momentrum gives Jfinal = 1. The spin S of the two neutron system is either 0 or 1. Thus for J = 1, we have two possibilities: Triplet, spin state (S = 1): 1' = 2 , 1 , 0 i.e. the final state is 3D1or 3P1 or 3S1. For the singlet spin state (5'= 0): 1' = 1 and the
Parity Constraints on S-Matrix for Hadronic Reactions
73
final state is 'PI. Now the Pauli exclusion principle requires that the final state must be antisymmetric. Since the triplet spin state is symmetric, the orbital state must be antisymmetric i.e. 1' = 1 and allowed final state is 3P1.For the spin singlet state, since it is antisymmetric, I' should be even. Thus 'P1 state is not allowed by the Pauli exclusion principle. Hence we have the result that the final state must be 3P1so that from Eq. (36), we get
qn- = (-1) 1' = -1 since r]d = +1. Thus for a pion doscalar particle. 3.4
Jp =
(3.37)
0- and it is called a pseu-
Parity Constraints on S-Matrix for Hadronic Reactions
3.4.1 Scattering of spin 0 particles o n spin particles Consider two-body elastic scattering of a spin 0 particle on a spin particle a
+
P1
b - + c + (P2, 4 Pi
d (P'z,4
In the center of mass frame
P1
= - P2 = p i
pi =
-
p'z = p z .
(3.38)
For the elastic scattering lpil = lpfl = IpI = p . The initial and final states can be labelled as I i ) = Ipi, 0 ) , I f ) = Ipf, 0 ). Under parity
p I4 = 17:
I-PZIU),
p If)
= lly l-Pfl
4.
(3.39)
The transition matrix elements
(3.40)
74
Space-Time Symmetries
Now invariance under P implies
P T Pt = T .
(3.41)
Because of elastic scattering
(3.42)
rliP -- rlf. P
Therefore, we have from Eqs. (40)-(42) k P f , 4 T I-Pi1
4 = ( P f l 4 T (Pi,0 ).
(3.43)
If we assume rotational invariance, then ( T ) can depend only on the rotat,ional invariant quantities p , pf . pi, cr . pi, u . p f , o . ( p ix p!) . We need not consider u2 or higher powers of it, because o2 = 3 and ( o . a)( u . b) = a , b i u . (a x b). Thus these quantities can be reduced to either a constant or o . In other words, assuming rotational invariance only, we can write in spin space
+
This is a 2 x 2 matrix in spin space. It is understood that the above matrix elements are to be taken between spin wave functions and xi for the final and initial states. Thus using rotational invariance alone, we have 22 = 4 independent amplitudes. If in addition we assume invariance under parity, then Eqs. (43) and (44) imply Al = 0 = Aa. Therefore, invariance under rotation and spaceinversion gives
x!
(Pf,01T l P i l 4 =
X: [ A(pi 0) + B ( P , 0) g *(pi x P ~ )xi. I
(3.45)
This is an example which shows how a symmetry principle restricts the form of a transition matrix.
Parity Constraints on S-Matrix for Hadronic Reactions
75
3.4.2 Decay of a s p i n @ particle into three spinless particles each
having odd parity Consider the decay A
+
+
p2 +p3,
where all the particles have spin 0. Consider the decay in the rest frame of particle A. We have
0 = P1+
P2
+ P3,
(3.46)
where p l , p2 and p3 are momenta of particles PI, P2 and P3 respectively. The transition matrix elements for the decay is given by
Under parity
Now
If parity is conserved PTP~=T
(3.50)
and we have from Eqs. (3.49) and (3.50)
M
(Pl,
P2, P3) = -M
(-P1, -Pz,
-p3).
(3.51)
Because of the rotational invariance, M can be a function of rota~ 1p1. (p2 x p3). tional invariant quantities p1 a p 2 , ~ 2 . ~ ~3 ,3 . and
Space-Time Symmetries
76
But the last invariant is zero, since p3 = - (p1 + p2). Hence the rotational and space-inversion invariance implies
A4
(P1 *
P2, P2 * p3, p3 ’ p1) = -A4 (p1 * p2, p 2 .p37 P 3 ’ P l )
or
M
= 0.
Thus we have the result, that the decay of a spinless particle with even parity to three pseudoscalar particles is forbidden if we assume invariance under space-inversion. On the other hand, decay of a spinless particle with odd parity to three pseudoscalar particles will be allowed under space-inversion invariance.
3.5 Time Reversal Under time reversal
t-b-t,
x-kx.
(3.52a)
Therefore, p+-p,
L-t-L,
u+-
0.
(3.52b)
Let Il denote the operation which transforms quant,iim mechanical states and operators under the above transformation i.e. under t ---t -t. First we show that II cannot be a unitary operator. Under IT, the commutation relation
(3.53) is not, invariant. Hence the transformation generated by I I cannot, be unitary. But we want the above commutation relation to be A way out of this difficulty is as follows: All invariant under c-numbers are simultaneously transformed into their complex conjugates. Such a transformation is called antiunitary. Then under
n.
n,
gi
-t
n Cz n-1 = &,
Fj
i
3
j
-i
n Fj n-1 = +jj
(3.54)
Time Reversal
77
and the commutation relation (53) remains invariant. Also, we note that II J n-' = -J (3.55) and the commutation relation [Ji, Jj]
(3.56)
= iEijk J k
is preserved. We now discuss the transformation of the transition matrix T under If HOand V are invariant under time reversal, then
n.
n-' = Ho n v n-' = v.
ll Ho
(3.57)
Now [cf. Eq. (2.55)]
l + ie v
T=-V-V
Ea- H
and we have
1 Ea - H - iT Invariance under time reversal implies
IIT
n-'=
-V-V
= (it
E
V =Tt.
IT I f t ) .
(3.58)
(3.59)
We now discuss the transformation of scattering states under time reversal. We specify a state I a ) by its momentum, zcomponent of spin ma, and a which denotes all other quantum numbers which may be necessary to specify the state. Thus we write
1
I4
=
I a , P a , ma)
)
=
I a, pa, ma)in
a(+)
(3.60)
SpaceTime Symmetries
78
Under
n n I
a , Par
ma)
=
10,
-Pa, -ma)
r
(3.61)
where we have used Eqs. (2.46), (2.47) and (57) and the rule i -+ -i. Let us specify the initial and final states as
(3.63) Then
I f">
=
I P,
-Pfr -mr)
(3.64)
*
Therefore, Eq. (59) gives
(A
I
Pi, mi)= ( a, -Pi, - m i v
I
-Pfr - T ) . (3.65) This expresses the equality of two scattering processes obtained by reversing the momenta and spin-components and interchanging the initial and final states. This is known as reciprocity relation and is a consequence of invariance under time reversal. Since ll is not a unitary operator, therefore, it does not have observable eigenvalues. The states cannot be labelled by such eigenvalues. Therefore, invariance under II cannot be tested by searching for time-parity forbidden decays. It can be tested by using the relation of the form given in Eq. (65). No violation of time reversal has been found in hadronic and electromagnetic interactions. P f , mfl T
Qr
Pl
79
Applications
3.6 Applications 3.6.1 Detailed balance principle Determination of spin of the pion
If we assume invariance under time reversal, we get Eq. (65). In addition, if we assume parity conservation, we have from Eq. (65). = =
(P, P f , T I T I Q, Pi, mi> ( Q, -pi, -mi1 F+TF 1 p, -pf, -mf) ( Q , Pi, - m i v I P, P f , - T ) . (3.66)
If the spins are summed, then we can mite
I Q, Pi, W)l2 P,
P f , mf)I2
(3.67)
spin
This is called the “semi detailed balance principle”. We now apply the above result to two-body scattering u+b-+c+d,
e.g.
Then we get [cf. Eq. (2.92)] d o -( u dR
+b + c +d)
and d o -( c + d dR
+a+b)
p+p+.rr++d.
Space-Time Symmetries
80
But Eq. (67) gives (3.70) spin
spin
Hence we have
da dS2
- (u
+b
3
c
+ d)
-
This is known as the principle of detailed balance. We now apply the above result, to the reaction pf p
-+ 7r+
+ d.
Then from Eq. (71), we get, da -(p+p-+7r++d) dC2
=
3 (2s,
4
+ 1)p:-do '.( + d p",Q
-+
p
+p ) , (3.72)
where we have used the result that the proton spin sp = and that the deuteron spin s d = 1. For the total cross sections, we get
From the experimentally measured cross sections, we find s, = 0 i.e. the spin of the pion is zero.
3.7
Unitarity Constraints
So far, assuming rotational invariance, we have discussed the constraints on the T-matrix imposed by space reflection and time reversal invariance. In this section, we discuss the constraints on the T-matrix due to the unitarity of the S-matrix.
81
Unitarity Constraints
Unitarity of the S-matrix gives
SS=1 or
IS S+ I
(3.74a)
I
(3.7413) where 1 i) and I j ) are initial and final states. Introduce a complete set of states I k ) , (j
c
(j
2)
=(j
IS I k ) ( k
2)
= sjz,
1st I i ) = S .3%.
(3.75a)
k
or
c
( j I [I
+
2
pn14s4
( P~ pk)T ] I k )
Ic
(ICI [I - i (aT14s4 ( P -~ pi)T'] I i) =
sji,
(3.75b)
which gives
-i ( 2 ~ ) ~[ s k i S4 (Pj - P k ) T j k
c s4 (p,
- Sjk
S4 (Pk - P Z ) (Tii)]
k
= (27r)8
-
pz)
1 k)
( j IT
( k 1 Tt
1 2) s4 ( R - Pk)
k
(3.76a) or
[
"31
-i T.. 2% - T.*. = (2~ )4
( j IT
I
k ) ( k ITt
I 2) s4
k
(e- Pk).
(3.7613) In Eq. (76b), C means integration over momenta and sum over k
other quantum numbers. Only those states contribute which are allowed by energy-momentum conservation implied by the 6-function in Eq. (76b). For forward elastic scattering and no spin flip, i = j and we get 21mzz = (
~C4(i I~T I IC) k
(IC
IT^ I i) s4 (
~ -k
a
(3.77)
Space-Time Symmetries
82
For two-body scattering viz. a+b-+l+2+.. the right-hand side of Eq. (77) is the transition rate Wi [cf. Eq. (2.69)J,where Wz(i = a 6 ) is given by
+
(3.78) Expressing the T-matrix, in terms of the amplitude F , we have (3.79a) where ma mb
a and b both fermions
a, EaEb
a boson, b fermion 1 a and b both bosons 4EaEb '
Hence, we have from Eq. (77):
1
.
(3.7913)
where Fii is the forward elastic scattering amplitude, ai = g a b is the total cross section for the reaction a b 3 1 + 2 + . . and
+
n, =
[ myb]
(3.81)
depending upon the nature of particles a and b. Equation (80) is known as the optical theorem. As a simple example, consider a and b to be spinless particles. Then we can express
83
Unitarity Constraints
If we put
fii (s,O) =
f(0) we have from Eqs. (80)-(82)
(3.83) the usual form of optical theorem in potential scattering.
Two-particle partial wave unitarity Assume that for each channel k, three or be neglected. We work in the center of state i = a b, so that pa = Pb = p. Two-body Lorentz invariant phase space
+
more particles states can mass frame, with initial w e take p along z-axis. is given by
In the center of mass frame, plk = - P 2 k = pk , where
(3.85)
N =
[
-1
both bosons 1st particle boson, 2nd one fermion mlk m2k7 both fermions 4’
y,
Then working out the integral (84), we get
1
. (3.86)
(3.87) where Q’ = (O’,4’) is the solid angle between p and pk. 0 = (O,$) is the solid angle between p and plj where plj is the momentum of first particle in the state j. a’’ = (O”,4”) is the solid angle between pk and plj . For the two-particle states in channel k, the unitarity relation (76b) becomes, on using Eq. (87)
Space-Time Symmetries
84
We use the general relation (88) for t,wo-particle unitarity for three important cases: Case (i): Collision between spinless particles. In this case i , j , and k are simply channel indices. For this case, we can expand Fij (8) in terms of the Legendre polynomials of cos0 [This is a consequence of rotational invariance; there can be no dependence on the magnetic quantum number rn and hence no dependence on $1, This expansion is given in Eq. (82). Similarly Fik: (0’)is independent of 4’ for spinless particles and can be expanded in terms of PL (cosf?’).Likewise Fjk (W) can be expanded in terms of PL (cosf?”).Hence, we have from Eq. (88) -28.n s1/2
(Fji, L (s) - FG,
(s))
(2L
L
- 1 x
Pk
+ 1) PL (cos6)
64r2 s
yx (2L”+ 1) (ZL‘ + 1)
$k,
(s)
F,*,
~r
(s)
I,!‘ L’
x
/ PLr/
(cos 0”) PL/(cos 0’) sin 8’ df?’ d#.
(3.89)
In order to evaluate the integral on the right-han~side of Eq. (891, we use the following formulae:
(3.90a)
(3.90b)
We get from Eq. (89), using Eqs. (90) 1 (2L 1) (Fji,L (4 -
2i
c
+
q,L (4) PL tcos8)
85
Unitarity Constraints
=
c c (2L’+ Pk
1)
$k,
L’
(s)
c k , L’
(s)
PL’(coso) .
L’
k
(3.91) Since the Legendre polynomials are linearly independent, we get the desired 2-body partial-wave unitarity relation 1
(4i, L (s)
-
q,L
(9))
=xpk
Fjk, L
q,L ( s ) .
(s)
k
(3.92)
If we are interested only in elastic scattering, we may drop indices i and j and we obtain ImFL (s) = x P k
I F k , L12
(3.93)
*
k
Occasionally all channels except the elastic one are closed at low energies. Then pk = p and we have
ImFL
= P lFLI2
,
!
(3.94)
so that we can put
1 FL= - ez6Lsin SL, P
(3.95)
where SL is a real function of s. We can also express
FL= - (e 2 i 6 L - 1) = p-1 (cot& - 2)-’ 2ip
(3.96)
The differential cross section is given by
(3.97) where we have used Eq. (82). Using the orthogonality of Legendre polynomials, we get
Space-Time Symmetries
86
where
+
P’ (2L 1) IFji,LI” (3.9813) P For “purely elastic” region, where Eq. (95) applies, we have Uj2,L
= 47T-
(3.99) Case (ii): Particles a and b carry spin. Here it is convenient to introduce helicity. Let X1 and X2 be helicities of particle a and b
respectively and let X = A1 - X2. In the center of mass frame pa = -pb = p. Let us take the vector p = ( p ,8,$) . In the center of mass frame we represent the two particles state as [st= (8,4)]
lp, h ,X2, st) = IP, X l ) I-P7
X2)
(-1) s2-x!2
(3.100)
The last factor in Eq. (100) is due to phase convention. Noting that (J = J1+Jz), J p = J l p - J2- (-p) , we have
(3.101) Now
R 10,O) = I@, 4)
(3.102)
where R is the rotation operator e-ien’J’h[n = (- sin 6, cos 4, O)] and
(3.103) where d&,M (0)are rotation matrices. Thus
a7
Unitarity Constraints =
1d L M f (R) ( J M‘ ,XI 0 0) M’
(3.104) Hence
le,@,X) =
c IJ
(J
M ,XlW
JM
=
5 /FlJ
M , A) d L x ( R ) .
(3.105)
Thus we can write
14 4, X1, X2)
=cIJ
2J+1 W 2 )
(Q)
(3.106)
JM
+
+
We now consider the scattering process a b = c d . Let A1 and X2 be initial helicities and Xl, and Xl, be final helicities. X = A 1 - A 2 and A’ = Xl, - Xl,. We take initial momentum p = ( p , 0,O) and final momentum p’= (p’,O,$). We can write the scattering amplitude on using Eq. (105)
x (J’M’
Al,Al,, j l F IJM
X1X2,i)
Space-Time Symmetries
88
81/2 Note that n = 87r2 m, mb m, md when all the particles are fermions. For spinless particles n = 32r2 sl/’, X = A’ = 0 , J = L and d$ (0,q5) = &Yi0 (0,d) = PL (COS 0) , we get back Eq. (82). The differential scattering cross section for the process a b + c d is given by
+
+
(3.108) where we have used
To proceed further we note the following properties of rotation matrices d;’x (-0) =
(R))A’A= (dJ’
I d ; ; ’ (R) d$,’ (0) dR =
= d;;’ ( Q )
47r
~ J J SAM ,
( 2 J + 1)
(3.110) (3.111)
d J (R”) = d J (-a’) d J (R) = d J t (R’) d J (0) or
diA,(0”) =
x M
( d J t (a’))AM . ( d J (R)) MA’
= Z d & ( O ’ ) dkA’(0).
(3.112)
M
Note that in Eq. (112) d J (R”) has been expressed as product of two rotation matrices corresponding to -R’ ,R. Then using Eqs. (107),
89
unitarity Constraints
(1lo)-( 112), we get from Eq. (92), for the two-particle partial wave unitarity relation. 1 2i
- [<: (XiX;; =
XJ2;
(Xi&;
F$
pk
s) - 4 ;
(A&;
XkiXle2;
s)
Xp;;
cr
s)]
(XlX2;
hciXle2;
s)
XklXk2
(3.113) The two-particle elastic unitarity gives 1 5 [ F J @A;; = p
c FJ
XJ2;
s)
(XiX;;XyX$
xl,x;
- FJ’
s)]
s ) F J * ( X , X 2 ; X ~ X ~ ; ~).(3.114)
p/y/ 1
2
Assuming parity conservation, we get
where ~ 1 , 2 3 2 ,si and 5’2 are the spins of particle a and b in the initial and final states and 7 is the product of their intrinsic parities. Equation (115) shows that not all the amplitudes are independent. Time reversal invariance puts additional restrictions on the amplitudes FJ’s namely
F; (XiXk;
X1X2;
s) = F;
(XJ2;
xix;;
.
(3.116)
s) = F J
(X1X2;
X i x i ; s) .
(3.117)
s)
For the elastic scattering:
F J (XiA;; A,&;
Finally using the orthogonality of d-matrices, we get integrated cross section for elastic scattering from Eq. (108)
Space-Time Symmetries
90
where
In particular, when all the particles have spin 1/2, the S-wave unitarity gives ( s = 4p2)
(3.119)
+
+
For special case for the elastic scattering of a b ---t a b, where a carries spin s and b is spinless, we have X = X1 - A2 = X I and A‘ = X i - Xl, = X i . For this case we have from Eqs. (107), (108), (117), (113) and (114) (for a to be fermion)
Fx/x
4 7 4
(n)= -C ( 2 J - t 1) F;A ma
(s)
d,Jlt
(0,d)
(3.120)
J
(3.122)
Fxix J --F J-x/-x.
(3.124)
We end this chapter with the following remarks. We have shown how the symmetry principles piit restrictions on the Smatrix. In this way, we get, the minimum set of observables to describe t,he experimental data. This approach is especially rewarding, when the underlying dynamics is not, known, which is the case for the hadronic interactions.
91
Problems
3.8 Problems
Problem 1: Nucleon-Nucleon Scattering Amplitudes
In the center of mass frame
Introduce three orthogonal unit vectors 1, m, n
1 = P x P'
IP x P'tl
lilj
+P IP' + PI
n = P'
m = P' - P IP' - PI
+ mimj + ninj = S,j
i
= 1,2,3.
Then T-matrix can be written as
It is understood that matrix elements are to be taken between the spin states X at f x bt f and xai X b i . Then using parity conservation and time reversal invariance, show that T can be written as
For identical nucleons like p
and n - n scattering, (2') has to be symmetric under 1 tf 2; hence HA = 0. Show that, -p
Space-Time Symmetries
92
By eliminating u l - n62.11, using the above relation, express (5") as
(T)
=
(z)
[ G+ ~ G~ n1 u2+ G~ 6
1 - m u2.m
show the most general form of 2-nucleon potential can be written in the form
where S l Z = Q12
=
3 U 1 . F a 2 . F - u1 u2 1 - [ C T ~ * L ~ 2 L .+ U Z L 2 *
*
d
= 0 1 +a2 =2s
L
= (rxp).
~
1
L].
Problem 2: Consider the elastic scattering a + b -+ a + b, where a is spin half particle and 6 is spinless. For this case J = L -f 1/2. Expressing the two independent amplitudes F:/z 1 / 2 and F&, as 1 J Fl/Z f 1 / 2 =
2 (h+* f ( L + l ) - )
where L* correspond to J = L k 1/2, and then using Eq. (123), show that 2 ImfL* = P IfL*l . Hence one can write
93
Problems fL- =
1
P
PL-
sinbL-
The scattering matrix [cf. Eq. (45)] can be written as
where n =IP
x pf
x P'I'
If p is along z-axis, then
xM, = e-ie 0 . n x M ,
n=(-sin$,cos$,O).
Using the relations (where the prime denotes differentiation with respect to cos O ) ,
show that
Now, using Eqs. (120) and (121), show that
94
Space-Time Symmetries
Bibliography
95
3.9 Bibliography 1. S. Gasirowicz, Elementary particle physics, Wiley, New York (1966). 2. H. M. Pilkuhn, Relativistic particle physics, Springer-Verlag, New York (1979). 3. T. D. Lee, Particle physics and introduction to field theory, Harwood, New York (1981).
Chapter 4 INTERNAL SYMMETRIES Hadrons found in nature are not fundamental constituents of matter. There are hundreds of them. They can be divided into two classes: (a) baryons: they are fermions with half integer spin i.e. J = 3/2,1/2; (b) mesons: they are bosons with integral spin i.e. J = 0,1,2. Some of the low lying mesons with J p = 0-and J p = 1- are shown in Figs. la, lb. Low lying baryons with J p = 1/2+, 3/2’are shown in Figs, 2a, 2b. Hadrons with the same J p are distinguished from each other by some internal quantum numbers. The assignment of these quantum numbers is meaningful, since these quan~umnumbers are additively conserved in hadronic interactions.
4.1 Selection Rules and Globally Conserved Quantum Numbers A particle would decay into two or more lighter ones if the decay is allowed by energy-momentum conservation. The reason is that the entropy S = kBln (phase space). Since phase space for the lightest particles is largest and the entropy S tends to increase, the system tends to decay into the lightest particles, unless there is some selection rule to forbid that decay. But we know that certain decays, although allowed by energy-momentum and angular momentum conservation, do not take place. Thus there must be selection rules or conservation laws which forbid these decays. We now list these “global” conservation laws: 97
Internal Symmetries
98
Mass (MeV) 960
rl'
s=o, I=O
550
*rl
K+S= 1. I=lG
4 94
S=-l,I=112 K-
140
7c-
Figure 1
s=o. I=O
;io Ko
0
71:
i-
n:
s=o. I = 1
Lowest lying pseudoscalar mesons (Jp = 0-).
Selection Rules and Globally Conserved Quantum Numbers
892- S=-1, I=1/2 $-
783 : 770
-
20
$0
P-
I=O
*+
K S=l. I=K
s=o, I = C
w0
-1 Figure 2
s=o,
0
1020-
99
+
Po
P S=O,I=l
0
i
Lowest lying vector mesons ( J p = 1-).
Q
100
Internal Symmetries
s= -2, I=1/2
1320
-0
1190
rp
111>
A0
938
n
p
0
1
Y Y
-1 Figure 3
c+s= -1,I=l s=-1; I=O
s=o, I=1/2
Lowest lying 1/2+ baryons.
Selection Rules and Globally Conserved Quantum Numbers
101
M (MeV)
1672-
fi
1530
Y L?-
s=-3,1 = 0
*fro
s = -2,
u
13 85
f-
f0
f+
123 2-
A-
A0
A+ A* S = 0; I = 312,
I = 112
S=-l.I=l
&
Figure 4
Lowest lying 3/2+ baryons.
102
Internal Symmetries
+
(i) Electric charge conservation: The decay e- + v y is not seen ( T ~> 4.3 x years). This is a consequence of electric charge conservation: “Electric charge is additively conserved in any process”. This in turn is a consequence of the invariance of Hamiltonian under the global gauge transformation: UQ (1) :
19)+ e2QA 19)
(4.la)
[Q,H ] = 0.
(4.lb)
so that,
The electric charge Q is a generator of UQ(1)global gauge group. If A is a function of space-time viz. A = A(r, t ) ,then the gauge transformation is called local. Actually, electric charge has a dual rule; it is also a generator of the local gauge group, U = eiQA(P,t) . It is a feature of local gauge group that corresponding to this transformation, there is a vector field A, coupled to the matter field 9, with a universal coupling whose strength is just, the electric charge of the particle represented by the field 9.None of the other quantum numbers has this feature. A closely related concept is the quantization of the electric charge, which at particle level is expressed as
i.e. the electric charge q of any hadron or lepton is an integral multiple of elementary charge e. In particular N, = 0 [qn = (-0.4f 1.1) x el and N, Np = 0 [l(qe + qp)l < 1.0 x el. (ii) Baryon charge c.onservat,ion: The following decays
+
p
-+
e++y
p -+ e+ + T O although allowed by electric charge conservation are not seen experimentally ( T ~> years). This can be understood, if we assign
Selection Rules and Globally Conserved Quantum Numbers
103
a baryon charge B as follows: +1 for baryons B = ( -1 for antibaryons 0 for leptons and mesons
(4.3)
and demand that B be additively conserved in any reaction
The corresponding global gauge transformation under which the Hamiltonian is invariant is given by
(iii) Lepton charge conservation: Some decay modes of leptons are not seen. The absence of these decay modes is a consequence of non-conservation of lepton charge which is assigned as follows: +1 for leptons L = { -1 for antileptons 0 for all other particles.
(4.6)
Any reaction in which L is additively conserved ( A L = 0) is allowed; oherwise it is forbidden. Some examples are given below: n + p + e - +
L O
L
fie -1 i;e
L -1
+ +
0 1 (2,A ) + 0 (2,A ) + 0
Allowed -1 A L = L f - Li = 0 ( 2 1,A ) e- not Allowed 0 1 AL=2 (2 - 1,A ) e+ Allowed 0 -1 A L = O fie
+
+ +
Further, the reaction [antineutrinos obtained from the decay of pile pee>] neutrons in a fission reactor ( n -+ p e-
+ +
Internal Symmetries
104
for which AL = 2 is not, seen, but the reaction using solar neutrinos,
v, +37 C1 --t e-
+37
Ar
has been seen and is allowed by lepton charge conservation. Also thc allowed reaction u,. + p + e+ + n has been observed with expected cross scction. The global gauge transformation, under which the Hamiltonian is invariant is given by IS) -+ *'ie I*). (4.7) It, was later discovered t,hat, tjhe neutrino produced in the decay n+ --t /-I+ u was not, the same as u, since if it were so, a reaction of the type
v + (2,A ) --t ( 2
+ 1, A ) + e-
would have been observed. Instead what, was observed was /-Ireplacing e- . This clearly shows that the neutrino accompanying p+ in n+ decay is different, from v, and is denoted by up. The muon number defined as $1 for /-I-) vp -1 for /-I+, ijp 0 for all other particles. is conserved in processes involving /-I*, v F , V p . (iv) Strangeness and Hypercharge: It is clear from Figs. 1-4, that, hadrons with the same spin and parity occur in nature as multiplets. Consider, for example, J p = 0mesons. We distinguish the triplet, of pions ( T * , T O ) , the doublets (K S , KO) and (I?' , K - ) by assigning a new qiiantiim number,
called strangeness: S ( n ) = 0, S ( K ) = +I and S (K) = -1. The singlets q and 7' have strangeness S = 0. Similarly the baryons with J p = 1/2+ are assigned the strangeness quantum
Selection Rules and Globally Conserved Quantum Numbers
105
number as follows: For the doublet ( p , n,),S = 0, for the triplet, (C*,CO), = -1, for the singlet (A'), S = - 1, and for the doublet, -0 ( z , ), S = -2. Sometimes, it is convenient to write Y = B S , where Y is called the hypercharge. The quantum number S is additively conserved in hadronic interactions. In any process, involving hadronic interactions, A S must be zero. This immediately leads to the result that in hadronic collisions, the strange particles are produced in pairs:
s -_ =
+
AS=O
+K-+C+ ---f
AS = -2
n + K+ + K -
A S = -1 A S = 0.
(4.8)
Experimentally, only the first and the last reactions are seen and the cross section for these reactions is typical of strong interactions. On the other hand, strange particles decay into ordinary particles by weak interact ions:
These decays have lifetimes of the order lo-'' seconds, characteristics of weak interactions. Thus strangeness is not conserved in weak interactions. In strong interactions, since both quantum numbers B and S are conserved, it is clear that hypercharge is also conserved. The gauge transformation under which the Hamiltonian is invariant is given by iPA lq,) IS) - + e (4.9) It is interesting to note that, the hypercharge of a mult,iplet,is just equal to twice the average charge of that multiplet, i.e.
Y = 2 ( Q )= 2
(q/e).
(4.10)
106
Internal Symmetries
For example for the triplet of pions ( T * , T O ) , (Q) = 0 and Y = 0, for the doublet ( p , n ) , (Q) = 1/2 and Y = 1, whereas for the doublet (I?', K - ) or (Z', Z-), (Q) = 1/2 and Y = -1. It is tempting to assign another quantum number, called isospin to each multiplet. For example we can assign I = l , I 3 = +1,0, -1 to the triplet, of pions (n+, T O , T - ) and I = 1/2, 13 = 1/2 and -1/2 to the doublet ( p ,n,). We will discuss isospin in the next section. Here we summarize the conservation laws for internal quantum numbers &, B , S and I . Interactions Quantum Number
Q B S or Y Isospin 4.2
Hadronic Electromagnetic Yes Yes Yes Yes Yes Yes Yes No
Isospin
We now introduce isospin. From Figs. 1 and 2, it, is apparent, that particles occur in nature as multiplets. In analogy with ordinary spin, we can regard proton and neutron as an isospin doublet, (nucleon) N
=
(g ) ,
with I = 1/2 and I3 = f 1 / 2 .
The concept of isospin is meaningful only if in hadronic interactions isospin is conserved. This is indeed the case. Experiments on nucleon-nucleon scattering show that after subtracting the effect of Coulomb force in p p scattering, p p , n p and nn hadronic forces are equal in strength and have the same range. That is nuclear forces do not depend on the charge of the particle and are thus charge independent. It, is now known that, all hadronic forces, not just the one between nucleons are charge independent. The two states of nucleon N viz. p and n will have similar properties as far as hadronic forces are concerned. Without electromagnetic interaction, proton and neutron will have the same mass,
107
Isospin
but its presence makes their masses slightly different. This is supported by the fact that (mp- m,) = -1.2 MeV only i.e. about 0.1% of mp. Like ordinary angular momentum, we introduce a quantity isospin I 3 ( I I , I 2 , 1 3 ) in isospin space. The operator i satisfies the commutation relations of angular momentum J viz. [fi, fj] = i ~ i j kfj, i =
1,2,3.
(4.11)
As a consequence of these commutation relations, it is possible to find a complete set of simultaneous eigenstates ( I 1 3 ) of f2, and 1 3 with eigenvalues 1(1+ 1) and 13:
13
has (21
+ 1) ) I 1 3 )
i2 ) I 1 3 )
= 1(I
13 [ I 13)
= 13 11 1 3 ) .
(4.12a) (4.12b)
+ 1) eigenvalues -I,
* * *
,+I.
(4.13a)
The possible eigenvalues of I are
1 = 0,1/2,1,3/2,2,
.
* *
(4.13b)
Thus, all the multiplets in Figs. 1 and 2 belong to an irreducible representation of the isospin group i.e. they have any of the possible eigenvalues of I given in Eq. (13b). For example, the proton and neutron states can be written as far as the isospin is concerned as
Id ) .1
=
IW 1 / 2 )
= 11/2 - 1/2)
>
(4.14)
and the pions can be represented as IT')
= 11 1)
ITO)
= (10)
In-)
=
11
-
1).
(4.15)
Internal Symmetries
108
The charge of a state is given by the relation
(4.16) This is called the Gell-Mann-Nishijima relation. Charge independence of hadronic force implies that this force does not distinguish any direction in isospin space that is to say that, hadronic interactions are invariant, under a rotation in isospin space in complete analogy with ordinary angular momentum. This means t,hat, the S-matrix or the hadronic part, of the Hamiltonian Hh commutes with the rotation operator
in isospin space i.e.
[ S , U I ] = 0, or
[Hh,
U,]
= 0.
(4.18)
i is the generator
of a rotation group in the isospin space. For an infinitesimal rotation
u,= 1 - za. I. I
(4.19)
Hence, we have
[s,i] = 0, or
i] = 0,
[ ~ h ,
(4.20)
i.e. isospin is conserved in any process involving hadronic interactions. Thus we have the selection rules
n 1q2= 0,
n 1 3 = 0.
(4.21)
Since in the absence of electromagnetic interaction, the mass Hamiltonian H M commiites with f , the eigenstat,es of H M with the same I , i.e. (21+1) states with different values of 13 , are degenerate in mass.
109
Isospin
7r
As an illustration of isospin conservation, we consider the - N scattering. 7T+p 7r-p
7r+p -+ 7r-p + 7r0 n,. --$
We can write IT+
p) = 11 1) (1/2 1/2) = 11 1/2 1 1/2)
IT-
p ) = (1 1/2 - 1 1/2)
pn)
=
(1 1/2 0 - 1 / 2 ) .
(4.22)
Now the scattering amplitude F is given by
(.-
=
P IF17.r- P )
cc(7r-
pl I’ I; 1 1/2)
113 I’IA
x (I’I; 1 1/2 IF( I 13 1 1/2) (I 13 1 1/22
=
C
(7~-
pl I - 1/2 1 1/2) FI (I
-
IT-
p)
1/2 1 1/2
IT-
p).
I
(4.23) Using the Clebsch-Gordon coefficients, we have (7T-
1 p(FI7r- p ) = - F$ 3
+ 23 F;. -
(4.24a)
Similarly, we get. (TO
Jz
Jz
n l ~ l n p- ) = - F? - - F~ 3 z 3 =
(4.24b)
Internal Symmetries
110
Without using isospin invariance we have three independent amplitudes. With its use we have only two independent amplitudes. Thus u,+
= p)9/
Ur-
E
2
[. p (7r-
(4.25a)
= u(+) -+ 7T-
p)
+u
(7r-
11
p -+ 7ro n, zz
J-)+ d o ) (4.25b)
Here p is the kinematical factor. If (24) u(+): u ( - ) :
F3/2
>> FIl2, then from Eq.
= g : 1 : 2.
Experimentally, the cross-sections are in the ratio (122f8) : (12.8f 1.10) : (25.6 f 1.3) for the kinetic energy of the pion from 120 MeV to 300 MeV. Thus it is clear that the scattering takes place predominantly in the I = 3/2 state for the above energy range. Finally, we note that since the electric charge is always conserved, the conservation of 1 3 implies Y-conservation and vice versa. To summarize, €or hadronic interactions
A(I(2 = 0 A(Q, B , Y ) = 0.
(4.26)
4.2.1 Electromagnetic interaction and isospin Because of Eq. (16), electromagnetic interaction breaks tthe rotational symmetry in the isospin space:
i3
[Hem, # O
(4.27a)
[Hem,f3] = 0.
(4.27b)
but Hence Hem is invariant under an isospin rotation about t,he 3rd axis i.e. 13 is still conserved by the electromagnetic interaction. We can say that the isospin symmetry is broken by the electromagnetic interaction and a small mass difference between the
Resonance Production
111
members of an isospin multiplet may arise due to the electromagnetic interaction. Since [Hem,
Q] =o,
(4.28)
therefore, it follows from Eq. (27b) that
Hence for electromagnetic interaction, we have the selection rules:
A13=0, but
AY=O,
AB=O
~ 1 1 1#~ 0.
(4.30a) (4.30b)
4.2.2 Weak interaction and isospin Consider the weak processes
A
+
n +
p+rp+e-+De.
Clearly 1 3 is not conserved in weak interactions and hence I2 is also not conserved. It follows that Y is also not conserved, since Q is conserved. Thus for weak interactions, we have the selection rules: AlII2 # 0, AY # 0 , AB = 0. (4.31)
4.3 Resonance Production We now consider the reaction shown in Fig. 5 . We have three particles in the final state, produced incoherently. Let us consider ) (T+ T - ) . We define the the pair of particles (nr+),( n ~and invariant mass of each system designated by (12), (13) and (23): s12
s13 s23
(El + E2I2 - (PI + P2I2 = (El + E3)2- (PI + p3)2 = (B2 + E3)2- (p2 + p3)2 =
(4.32a) (4.32b) (4.32~)
112
Internal Symmetries
Figure 5 The reaction
7r-p
--+
~ , T + T - ,r
-p
.+ p7r07r-.
Figure 6 The pion production through resonance T-P nm+T - .
+
A+T-
+
Resonance Production
113
If the reaction proceeds as in Fig. 5 , n,, 7r+ and rr- will have energy and momentum statistically distributed. The number of ( n n+) pairs with an invariant mass N(s12) can also be caland the result culated. N ( s l z )can be plotted as a function of is called a phase space spectrum as shown in Fig. 7. If the reaction takes place as shown in Fig. 6 , i.e. with T + ' S strongly correlated with the n's, then energy-momentum conservation demands
a,
EA
=
El
6
+ Ez
PA
= P1 +P2
ma
=
[Ei-pa]
112
=&.
(4.33)
In this case the final n T+ results from the decay of a quasi-stable particle A+, called a resonance. In this situation, N(sl2) shows a strong peak at& = VIA (Fig. 7). The finite width of the peak shows that the particle is very short lived, the life time 7 = $, r being the width of the resonance. Actually a broad peak is seen = VIA = 1238 MeV with the full width at experimentally at half maximum x 120 MeV. Similarly, if we consider the pair n T - , one finds a peak due to A-. The (7r+n-)invariant mass distribution, N ( s 2 3 ) , also shows a broad peak at about , & = 750 MeV, due to the po resonance.
6
A-resonance We now discuss the quantum numbers of the A-resonance. We first determine its isospin. The resonance A is seen both in rr-p and n+p scattering. Since for 7r+p, I = 3/2 is the only possibility, it follows that its isospin must be 3/2. This is confirmed in the 7rsp and n-p scattering experiments at energies at which multiple mesons production is insignificant viz. the processes:
If the I
T+p
+
7r+p
7r-p
-+
7r-p
-+
7r
0
n.
= 3/2 channel dominates in the above processes, we then
114
Internal Symmetries
1000
1400
1200 $2
Figure 7
Phase space plot for (m+)pairs.
Resonance Production
115
170 -
180
160 150 -
130 120 s 110 140
-
-
2
90-
E
*loo
8070
-
6050
-
40
-
a+ , '4
30 : 20
-p SI
101 0
gm
Figure 8
y
1100
The resonance scattering for ~ + and p T-P channels.
Internal Symmetries
116
have from Eq. (25) ~ ~ + = 3,/ at ~ the resonance ~ energy. This is what is borne out experimentally, showing iinambiguously that, the resonance channel is I = 3/2 (see Fig. 8). Spin of A We first consider two-body s(~at,terir~g
a + b -+ R
-+
a‘+ b‘
(4.34)
through a resonance R. Suppose the spin of R is J . Consider the decay
R--+a+b. Let, p be the momentum in the center of mass frame of particles a and b. Let XI and A:! be t,heir helicities. Now jp/ = p and its direction is given by w (B, 4). We can write the helicit,y state (cf. Eq. (3.105)]
=
1x1 A2
4=
c
2 J’ ~
+1
*n dLA (B,4) ~ ~ ’ ~ ‘ ,(4.35) ~ } f
J‘M‘
where
Now
X = XI - Xz. Therefore, the decay amplitiidc is given by
Resonance Production
117
or
(4.40) where we have used the orthogonality of d-functions. When R, a and b all are bosons, we get
(4.41) For a resonance scattering as in Eq. (34), the invariant scattering amplitude is given by
F(ab + R
3
a’b’) = x F ( a b --+
R)F ( R .+ a’b’) $ R ( s ) , (4.42)
M
where $ R ( s ) is the resonance factor. Now using Eq. (38), we have
F(ab
-+
R
-+
a‘b’) =
c
d$&’)
d;l;A/(wN)( 2 J + 1)
M XFf
(ab
-+
R ) F; ( R -+ db’)
$R(S),
(4.43)
where w‘ = (S‘,$’) and w” = (B”,$”) are the polar and azimuthal angles of particles a and a’ with respect to some fixed direction. Using the group property of d-functions
where B and $ are the polar and azimuthal angles of the particle a‘ relative to a. Hence we have
Internal Symmetries
118
Now comparing it, with [cf. Eq. (3.120) for the
Ja partial wave]
we have
Now the partial wave cross section in the angular momentum state J is given by
Using
2
( p i (ab 3 R ) (
2
=
IF; ( R -+ ab)(
(4.49)
and Eqs. (40), (47) and (48), we get gJ =
4rr
2J+1
2 JpI (2SU + 1) (2Sb
+ 1)
[r ( R
--t
ab)
r ( R+ d b r ) 4
Ih(S)l2]
(4.50)
The resonance factor is given in the Breit-Wigner form:
(4.51)
.rr+p -+ A”
3
r’p.
(4.53)
Resonance Production
119
From Eq. (52), we get
Experimentally, near the resonance
(4.54) giving J = 3/2. It is also possible to determine the spin of a resonance by angular distribution of its decay products. This we illustrate by considering the A-resonance viz. A++ -+ r + p . Take the z-axis along the direction of the nucleon (or pion) in their center of mass frame, so that 1; = 0 (2-refers to r t p in the initial state and 1 refers to orbital angular momentum). Since pion is spinless, M i= f 1 / 2 . If J is the spin of A-resonance, then M = f 1 / 2 , by angular momentum conservation, Now from Eq. (39b), the angular distribut,ion of pn+ in the final state is given by
(4.55)
(4.57) Thus the angular distribution is isotropic. For J = 3/2 and M = f 1 / 2 , we have
wa
+
((F;/;(Sjl2
/~3;:~(4~) (ld;;; 1 / 2 ( ” ) 1 2
+ ld;;;
- 1 , 2 ~ 0 ) / 2. ) (4.58)
Internal Symmetries
120
Again using [Problem 3.21, we have I ( 0 ) o( (1 + 3cos
.
(4.59)
We note that, I ( - 0 ) = I(0). The observed angular distribution of the protons or the pions at, the resonance agrees with the prediction of Eq. (59), showing that ,I = 3/2 for t,hc A. The above derivation clearly shows that, t,he angiilar distribution depends only on the value of J , and not, on the parity i.e. orbital angular momenturn which never enters in the helicity representfation used above. 4.4
Charge Conjugation
It is a general feature of relativistic quantum mechanics that corresponding to a particle, there is an antiparticle which has the same mass and spin as it,s particle. We tmat, particle and antiparticle on equal footing. We, therefore, postulate an operator U,, which changes a particle into its antipart
i.')
uc uc
=
IP) =
I.->
(4.60~~)
I$.
(4.60b)
In general, for a charged particle u c
I&,
P,
S> =
I-&,
P,
S>
(4.61)
where IQ, p, s ) represents a single particle stake with charge Q, momentum p and spin s . Now
I&,
Q IQ, uc Q IQ,
P, P,
S)
=
Q
S)
=
Q I-Q,
Q u,I&,
P,
S)
= =
P, s ) P, s>
dj I-Q, P , S) -& I-(2, PI s ) .
(4.62a) (4.62b) (4.62~)
Therefore, we have
u c s + s u c=
0,
(4.63a)
[ucQ]+ =
0,
(4.6313)
Charge Conjugation
121
i.e. U, and Q do not commute. Hence it is not possible to find simultaneous eigenstates of U, and Q. In general, for any additive internal quantum number, such as Q, 13, B , Y and L , uc
I&,
1 3 , B , Y, L )
=I-&,
-13> -B, by, -L)
(4.64)
[uc, Qil
# 0,
(4.65)
and consequently, where
Qi = f 3 , B ,
Y , or
i.
Now UC
u,"
B) B)
I-B) = u, I-B) = IB). =
Therefore,
u,2 = 1
(4.66) (4.67)
and eigenvalues of U, re f l , i.e. U, is a discrete transformation. It follows from Eq. (64) that states with Q # 0, B # 0, Y # 0, etc. cannot be eigenstates of U,. Only states with Q = 0, B = 0, Y = 0, 13 = 0 can be eigenstates of U,. For them it is possible to define the charge conjugation parity 17,:
uc la = O} = q c
IB = O ) ,
(4.68)
where 7: = 1
or qc =
(4.69)
qc is a multiplicatively conserved quantum number in any process which conserves C-parity. The C-parity is either +1 or -1. Charge conjugation is an internal symmetry. If
[U,, H ] = 0,
or
[U,, S ] = 0,
(4.70)
we say that the corresponding interaction is invariant under charge conjugation U,. While strong and electromagnetic interactions are invariant under U,, weak interactions are not [uc,
Hweak]
#0.
(4.71)
122
Figure 9
Internal Symmetries
The neutrino with helicity -1 and antineutrino with helicity
+l.
This is clear from the fact, that, neutrinos and antineutrinos which come out in P-decay of nuclei have opposite polarizations or helicities [ H = 2s - p/ lpl] . If charge conjugation were conserved in weak interactions, neutrino and antineutrino would have the same helici ty. How to test charge conjugation in hadronic interactions? Consider for example, the reactions p+p
+
7r++h
-+
7r-
+h,
where h (6,) denote all other hadrons with B = 0 and with positive (negative) electric charge. Now (pp
IS1IT+ h )
= (pp
lUF1 U,S UT' Ucl 7~' h)
= (ppISV-
h),
where we have assumed that S is invariant, under
u, s UT1 = s.
(4.72)
U,: (4.73)
123
Charge Conjugation
Thus C-invariance requires that posit,ive and negative pions have the same energy spectrum. Comparison of 7r+ and T- distributions show no difference, the result is stated as
C - nonconserving amplitude 5 0.01. C - conserving amplitude As we have discussed, y, 7ro and qo can be eigenstates of U,. We now determine the C-parity of these states. Now under U,, the electromagnetic current jEm:
(4.74) But the electromagnetic field A, satisfies the equation
Thus from Eq. (75)) it follows that
A,
UC
+-
A,.
(4.76)
Since a photon is a quantum of electromagnetic field, it, follows that, the C-parity of photon is -1 viz. q c (7)= -1.
(4.77)
The decays T O + 2y and qo + 27 have been observed. Hence if these reactions proceed via electromagnetic interaction, it then follows from C-conservation that qc
(TO) =
+1
q c (qo) = +1.
(4.78)
Since 7r" + 37 and qo -+ 37 can proceed via electromagnetic interaction, but have never been seen, these decays are strictly forbidden
124
Internal Symmetries
due to C-conservation in electromagnetic interaction. We conclude that the electromagnetic interaction is invariant, under U,.
Consider now the positronium, the bound states of e- and e+. Let us consider e- - e+ in definite (1, s ) state. Now e- and e+ are identical fermions which differ only in their electric charges. We can use a generalized Pauli principle for the positronium viz. "under total exchange of particles (which consists of changing simultaneously Q, r and s labels), the state should change sign or be antisymmetric". Under exchange of space co-ordinates, we get a fact,or (-l)', under spin co-ordinate exchange, we get a factor (-l)'+' ( s = 0 for spin singlet state and s = 1, for spin triplet state), exchange of electric charge gives a factor qc. We require the state to be antisymmetric, i.e. (-1)l
(-,)'+I
qc = -1
(4.80)
or qc = (-1)l+"
(4.81)
which gives the charge conjugation parity of t,he positronium in (1, s) state. The positronium (e- - e+) can decay into R y by electromagnetic interaction. C-parity conservation gives
(-1)l+" = (-1y.
(4.82)
From Eq. (82), we get, the following selection rules: z=o=s
Allowed
s=l
Similarly for ( p - p ) and quark-antiquark systems: qc = (-l)'+'.
G-Parity
125
Now for (T+ - T - ) system for which B = 0, Y = 0,Q = 0, generalized Pauli principle requires that the state should be symmetric (even) under total exchange of pions that, is (-1)l
qc = 1
(4.83)
or qJc= (-1)'.
(4.84)
Similarly for T O - T O system we get, qc = (-1)'. For this case, since two TO'S are identical particle, ordinary Pauli principle requires that (-1)' = even i.e. they must be in an orbital state with 1 even. Thus qc must be +1 for ?yo - T O system, whereas qc depends upon I value for T + T - system. 4.5
G-Parity
For strong interactions, both isospin and C-parity are conserved. For hadrons, it is convenient to define a new operator G = charge conjugation +180" rotation around 2nd axis in isospin space. It follows that strong interactions are invariant under G, but
i.e. electromagnetic and weak interactions are not invariant under
G. Under 180" rotation around the 2nd axis in isospin space, we have
(4.86) Therefore, we get lT*VO)
+-
p).
(4.87)
Internal Symmetries
126
Under charge conjugation
1.')
I% 3 ITO) .
T') TO)
(4.88a) (4.88b)
Thus we have (4.89a)
- IT'>
IT")
and ITO)
3-
ITO)
.
(4.89b)
Thus the G-parity of pions is G ( T )= -1. The nucleon state IN), under 180" rotation about 2nd axis in isospin space transforms as (NR) ==
,a
T2
TI2 T
IN)
+
(cos - i 2 = ir2 IN) =
T2
sin T 2
)
IN) (4.90)
i.e.
But
9IF) In,) 9in). lP)
(4.92)
Therefore,
-5 In> In) -5 - 113). lP)
(4.93a) (4.93b)
Only states with B = 0 and Y = 0 for which isospin I is integer can be eigenstates of G. Only for such states we can define G-parity G. In general G-parity of a state with isospin I is given by G=qc(-l) I . (4.94)
Problems
127
Thus for fermion-antifermion system, the G-parity is given by
G = (-l)l+s+' For
(T+T-)
system G=
4.6
(4.95)
= rlc (-1)I.
(-1)' = (-I)'+' = 1.
(4.96)
Problems 1 Consider pion nucleon scattering
+N + +N
Ti
7rj
where i and j are isospin indices of the incoming and outgoing pions respectively. Using isospin invariance, show that in isospin space, the scattering amplitude A can be written
Show that isospin
and
projection operators are given by
2 f t . r p3/2 =
PI12
=
1-t.7 3
3 ' where T are Pauli matrices and t are isospin matrices for I = 1. Further show that
Hint:
(t&
= i€j&
2 Show that for the decay
i+f+r, either AI = 0 or lAI( = 1.
Internal Symmetries
128
Hence show that for the decay q 3 7r+7r-y, pions are in I = 1 state and 1 is odd, but, for the decay w ---t 7rr+n-y,pions are in I = 0 or 2 state and 1 is even.
3 Show that, the decay w + 7r+7r- is forbidden in strong interaction, but is allowed by electromagnetic interaction. What are the valiies of isospin I and orbital angular momentum for the pions ?
4 Show that q
--+ 7r+7r-7ro is forbidden in strong interaction but, is allowed by electromagnetic interaction. Determine the possible vahies of isospin for the final pions.
5 Derive Eq. (94).
129
Bibliography
4.7 Bibliography 1. S. Gasirowicz, Elementary particle physics, Wiley, New York (1966) 2. H. M. Pilkuhn, Relativistic particle physics, Springer-Verlag, New York (1979). 3. T. D. Lee, Particle physics and introduction to field theory, Harwood, New York (1981). 4. Particle Data Group, The European Physical Journal C3, 1 (1998). I
Chapter 5
UNITARY GROUPS AND SU(3) 5.1
Unitary Groups and SU(3)
Consider a vector 4i, i = 1,2,. - - , N in an N-dimensional vector space. An arbitrary transformation in this space is
For a unitary group U ( N ) in N dimensions,
For the group SU(N), we also have det a = 1.
(5.3)
The basic assumption of all the group theoretical approaches to classification of hadrons is that particles belong to an irreducible representation of some group (in our case S U ( N ) )and form a multiplet and thus have the same space-time properties, especially the mass, spin and parity. The basic mathematical problem is the investigation of the representations of a group. There are two approaches to this investigation (i) global way, (ii) infinitesimal way. For continuous groups it is convenient to restrict to (ii). For the general infinitesimal transformation:
Unitary Groups and SU(3)
132
Then conditions (2) and (3) give =
-E;
= 0.
&:
(5.5)
The unitary transformation corresponding to (4)may be written as U ( u ) = 1 - A; 0 ( E ~ ). (54
~i+
A; are called the generators of the group U ( N ) and characterize the group completely. The N x N unitary complex matrices U ( n ) form the representation of U ( N ) . Hence there are N 2 arbitrary real parameters and thus there are N 2 generators of the group U ( N ) . For SU( N ) we have N 2- 1 generators because of the unimodularity condition. The matrices U (a ) have tlhe group property:
U(b) U(a) = U(C) U ( a ) = U ( a ) U ( l ) , U(1) = 1
u-l ( b ) U ( U ) U ( b )
=
U(b-'ab)
U +( u ) U ( a )
=
1.
(5.7)
It is easy to see that Eqs. (5) and (6) give t
(A;)
= A'.
(5.8)
By taking a tlo be infinitesimal transformation, it, is easy to derive (see problem 5) the commutation relations
[A:, A:]
=
6; A: - Sf A;.
For the transformation (4),we have
(5.9)
Unitary Groups and SU(3)
133
But we can write
(5.11) Comparing Eqs. (4)and (ll),we get (5.12) Hence from Eq. (10) we have
=
6;6;
$1
= 6; 4 j .
(5.13)
[
t The matrices Mj ( M j ) = M!] give the representation of the group
U ( N ) for the fundamental representation
4i. We define a vector
8 = 4'. ItJ belongs to the representation fl of U ( N ) , whereas vector q$ belongs to the representation N of U ( N ) . Thus q3i transforms as
4
qyi
- j
=
4;
=
(@-.ti)
4;
(6; -.;)@.
(5.14)
Hence it follows that 3
Now if we consider a tensor get
= -6;
8.
(5.15)
T,'", it transforms as 4k#l, so that, we
[ A f , T k ]= 6: T: - 6;
q,
(5.16)
Thus the tensor T' transforms in the same way as the generator
A;.
Unitary Groups and SU(3)
134
Let us now restrict ourselves to S U ( N ) . The generators of S U ( N ) must be traceless. Hence we can write its generators as
so that
U(U)
=
1 - E: Fi
(F;)+ = F:
q
=
0.
(5.18)
Since A; is a U ( N ) invariant, the commutation relation for F' remains the same as in Eq. (9) viz.
[F, 3 qk] = 6; FJ - 6; F / .
(5.19)
The matrices Ad; must now be traceless, hence we have
(5.20)
Thus we have instead of Eqs. (13) and (15),
We now confine to S U ( 3 ) . It, is convenient, to express eight generators F', ( i , j = 1 , 2 , 3 ) in terms of hermitian operators FA, ( A = 1, . . ,8) introduced by Gell-Mann. The relationship between FA and F; is as follows:
Fi
=
Fl-iF2,
F?=Fl+iF2,
Fl
=
Fd-iFS,
Ff=F*+iFs,
Fi
=
F6 - i
F3 - --Fs,
2
F7,
3 -
1 -(F';-F;)=F3, 2 F:=F6-iF7,
Unitary Groups and SU(3)
135
From the commutation relation (19) for Fj, one can show that FA’S satisfy the standard commutation relation of a Lie group:
where the structure constants fABC are real and antisymmetric. FA’Salso satisfy the Jacobi identity
Infinitesimal unitary transformation generated by FA is
U = 1 - i E A FA, being infinitesimal real parameters. For an inifinitesinal transformation, the vectors 4iand @ transform as EA
since the matrices XA are hermitian. The matrices XA are related to Mj in the same way as FA are related to Fi . Thus
1
A.4;= - (A, - 2 x 2 ) , M? 2
=
1 - (A1 2
+ i&),
*
a ,
M;
=
1
--&.
d3
(5.26) Now for S U ( 3 ) ,the matrix elements of matrices Mj are given by (5.27)
Unitary Groups and SU(3)
136
Using Eqs. (16) and (17), we can explicitly write 3 x 3 matrices X A . They are A1
=
( 10 01 0 ) , X 2 = ( 20 -20 0 0 0
0
0 0 1
0 0
0 0
0
O0 ) , h = ( O 1 -10
0 0 0 0 0 0
0 -2
0
0
0).
0
(5.28) 0 2
0
0
0
-2
satisfy the same commutation relations Obviously the matrices as the generators FA, so that [XAI XB]
=2i
f A B C XC.
(5.29a)
They are traceless and have the following properties:
[ X A , XB]+
= 2 dABC
4 XC
-k j 6 A B ,
(5.30)
where d A B C are real and are totally symmetric. Defining = 8 1 . the commutation and anticommiitation relations can be written as
where A , B , C = 0 , 1 , . - . , 8. Thus XA are closed both under commutation and anticommutation. We also note that Xz, As, A7 are
Particle Representations in Flavor SU(3)
137
antisymmetric while the rest of them are symmetric. We express this fact by writing
A: = V A XA
(not summed),
(5.32)
where V A = -1, for A = 2 , 5 , 7 and +1 otherwise. The following identities follow from Eqs. (31) and (32): V A ?B VC f A B C
= -fABC
V A TIB Vc d m c =
dABc
(repeated indices not summed) (5.33)
i.e. ~ A B C( ~ A B C )is zero if even (odd) number of indices take t,he value 2, 5 or 7. The values of ~ A B Cand ~ A B Chave been tabulated by Gell-Mann and are reproduced in Table 1. The role of FA is the same in S U ( 3 ) as that of isospin I in S U ( 2 ) and for this reason FA'S are sometimes called component of F-spin. 5.2
Particle Representations in Flavor SU(3)
Out of the eight tensor generators F'. of SU(3), the set F: F i Ff and F: form the generators of the subgroup S U ( 2 ) x U(1). We have SU(3) 3 S U ( 2 ) x U(1) 3 S U ( 2 ) . It is convenient to classify states in an S U ( 3 ) representation by making use of this fact. The generators of the S U ( 2 ) x U(1) subgroup which are conveniently taken to correspond to isospin and hypercharge are
I+ = F,2, I- = F;, 13 = -1 (F; - F,) 2 2 Y = F,'+F,2 = -F&
(5.34)
in the case of S U ( 3 ) group. There are thus two diagonal operators in S U ( 3 ) , namely Is and Y * SU(3) is, therefore, a group of rank 2. Further if we define the electric charge as
Q = F: in S U ( ~ ) ,
(5.35)
138
Unitary Groups and SU(3)
Table 5.1 Values of
ABC
fABc
ABC
~ A B C
123 147 156 246 257 345 367 458 678
1 1/2 -1/2 1/2 1/2 112 -112 fij2 fl/2
118 146 157 228 247 256 338 344 355 366 377
114 112 1/2
1/f1 - 1/2
112
1/f1 112 1/2 - 112 - 1/2
888 doAB
&GhAo
fABC
and
~ABC
Particle Representations in Flavor SU(3)
139
Eq. (24) give the Gell-Mann-Nishijima relation
Y &=Is+-.
(5.36)
2
The fundamental representation is a vector which we write as qi. Let us take (5.37)
as the field operator which creates a u-quark, or a d-quark or an s-quark viz.
u 10) = I.),
d[O)= Id) , s (0) = Is) .
(5.38)
The field operators qi belong to the representation 3 of SU(3), whereas the field operators
[ =[ 1
qi=qzf=
U
f ]
(5.39)
belong to the representation 3 of SU(3). qi create antiquarks or annihilate quarks. From Eq. (21), we have
In the matrix notation, we can write the field operators as row and column matrix respectively viz.
q = (G =
qi
and qi
ds)
(I).
(5.41)
140
Unitary Groups and SU(3)
t'
Figure 1 Weight diagram for 3.
Then it, follows from Eqs. (24) and (25):
(5.42) Hence we see from Eqs. (34), (36), (40) or (42), that the quark states or simply quarks belong to the triplet, representation of SU(3) and have the following quantum numbers:
I
Y 1/3
-l/2 0
-2f3
Q 2/3
= 1/2
Id) Is)
1/2
13
I4 I=O
1/3
-1/3 -1/3
It is convenient to plot each state of the triplet representation on an I - Y plot as shown in Fig. 1. Such a diagram is called the weight diagram. The 3 representation of SU(3) is not equivalent to 3; it transforms as qz = q t . It is the hypercharge which distinguishes 3 and 3. Antiquarks belong to the 3 representation of SU(3) and the weight diagram is shown in Fig. 2.
Particle Representations in Flavor SU(3)
Figure 2
141
Weight diagram for 3.
Mesons: Quarks are taken to be spin 1/2 particles. To build observed particles from quarks, it is convenient to assign a baryon number 1/3 to quarks. Thus
Consider
P," can be regarded as a field operator for pseudoscalar mesons. Thus ' z k q k ) 10) (5.43) = (qiqj - -6jq (0) (q) 3 has baryon number zero and is an octet. It may be taken to represent octet of pseudoscalar mesons T , K and 7. We write (in our
Pi
142 Table 5.2
Unitary Groups and SU(3)
Pseudoscalar Mesons J p = 0- [cf. Eqs. (43) and
(44)]
D in the last column denotes the dimension of the S U ( 3 ) representation, in this case 8. The negative signs in front, of certain states appear because of our phase convention. notation upper index is row index and lower index is column index)
(5.44) where identification is shown in Table 2. Hence in a matrix notation, the pseudoscalar mesons J p = 0- can be represented by a matrix:
The singlet pseudoscalar meson ql is given by
Particle Representations in Flavor SU(3)
Figure 3
143
Weight diagram for pseudoscalar meson octet.
Another possible set of candidates for the octet of bosons is vector mesons J p = 1- :
A singlet vector boson is denoted as wl. In broken S U ( 3 ) , a singlet meson can mix with the eighth component of an octet. For example, w8 and w1 can mix and physical particles are mixtures of them and are denoted by w and $. The weight diagram for mesons is given in Fig. 3. Baryons: We now consider baryons. Baryons have B = 1 and they must be constructed out of 3 quarks. For this purpose we proceed as follows. We write q j qk
=
1
2 (qj
qk
-k qk qj) -k
51 ( q j q k - q k q j )
Unitary Groups and SU(3)
144
where the symmetric tensor
(5.47) has six independent components. The antisymmetric tensor 1
has three independent components. Now a vector Tibelonging to the representation 3 can be written in terms of Al, as
(5.49) Hence we have the result
and 3@ 3@ 3 We first consider
= (6 C3 3)
+ (3 8 3)
3 63 3 :
viz.
3 @ 3 = 8 @1. We write the octet operator for baryons as
(5.5 l a )
Particle Representations in Flavor SU(3)
145
where (5.51b) For the singlet representation, we have
(5.52) We now consider 6 8 3: It is given by
where p{ijk)
=sij
qk
+
s j k qi
+ s k i qj
(5.5313)
is complet,ely symmetric tensor and has 10 independent, components. Now we show that,
Hence from Eqs. (53) and (54), we get
Unitary Groups and SU(3)
146
Hence we have 6@3=10@8
We write the decuplet, representation:
(5.56) and the octet (8’) representation:
B;
0.
=
(5.57)
Hence finally we have the result, 3@3@3 = (6@3)@3 = =
(6@3)@(3@3) 10@8/@8@1.
Baryon States: (i) Octet Representation 8: From Eqs. (51), we have
Bj (0)
=
IBj j
and for representation 8’ [cf. Eq.(57)]: ikl
(5.59)
Particle Representations in Flavor SU(3)
147
Table 5.3 Baryons J p = !j+
1
y 1 1 0
1
0
0
-1
1
-1
0
0
0
0
0
&
I
1 0 1
; ; 1
0
-1
0
I3 1 2 -1 2
1/2 -1/2 1/2 1/2
-1 -1
Unitary Groups a n d SU(3)
148
The octet of baryons are then identified as given in Table 3. Hence from Eq. (58) and Table 3, we see that known eight, J p = 1/2+ baryons can be represented as 3 x 3 matrices
[
B!3
c+
$A0;-LCoJz
L A O - LEO
--
6
u
-.
B!3
E$
n,
Jz
u
(5.60a)
c-
L A O - L-p 6 - Jz
=
n,
c
2 20
).
- LA0
4
(5.60b) Note that
B;
=
B'j yo,
(5.61) where the symbol * denotes complex conjugation with respect to SU(3) but hermitian conjugation for the field operators. The weight, diagram for the octet, representation is shown in Fig. 4. (i) Singlet representation 1: From Eq. (52), we have
-
1 J[d, S]
fi
U+
[s,
U]
d + [u, d ]
(ii) Decuplet representation 10: From Eq. (56), we have 1
IT.. ) z3k
= - { SZJ. .q k f
J3
sjk
qi
+ski qj}
.
(5.62)
10) .
(5.63)
S)
The detailed identification of the states of decuplet, representation are given in Table 4. The weight, diagram for the decuplett of baryons is shown in Fig. 5.
Particle Representations in Flavor SU(3)
149
Y n
1
L g
Figure 4
Figure 5
-1
Weight diagram for
Weight diagram for
++
baryon octet.
;+ baryon decuplet.
Unitary Groups and SU(3)
150
Table 5.4 Decuplet J p = 3/2+ [cf. Eq. (63)]
Quark Content Iu4
1l u d u + d u u + u u d ) d3 J- u d d + d d u + d u d ) 41 ldd4 1 .J u u s + u s u + suu)
LT 43
+sdu uds
++dus sud++dsu usd )
3 1s d d + d d s + & d )
r J.3 l u s s +
ssu+ sus)
-L fi (&s+
ssd+ sds) ISS4
Q I I 3 Y 2 ; ; 1
2:3
-12 1 -3 2
1 1 1
1
1
0
0
1
0
0
-1 0
1
-1
1
-1
0 -1 -1 -2
1 0 -1
53
1
-1 -1
; 5
0
f 2 0
Irreducible Representations of SU(3)
5.3
151
U-Spin
We have labelled the states within an irreducible representation of S U ( 3 ) uniquely by the eigenvalues of 12, I3 and Y .The reason is that S U ( 3 ) contains the direct product of S U ( 2 ) I x U ( 1 ) y i.e.
S U ( 3 ) 3 SU(2)I x U ( 1 ) y . The generators of S U ( 2 ) I and U (l ) y are identified with the generators of SU(3) as given in Eq. (34). However we can take a different decomposition. For example the generators
(5.64) are the generators of group SU(2)u . These generators commute with the generator Q = F:. (5.65) Thus we can decompose S U ( 3 ) as follows
S U ( 3 ) 3 SU(2)u x U ( 1 ) Q . Therefore, it is possible to label the states within an irreducible representation S U ( 3 ) by the eigenvalues of U 2 , U3 and Q , The generators of SU(2)v commutes with the generator Q = F:, thus U-spin is very useful when dealing with electromagnetic interactions. Just as each isospin multiplet is associated with a definite hypercharge, each U-spin multiplet has a definite charge.
5.4
Irreducible Representations of SU(3)
We have already encountered two irreducible representations: triplet = qi ' i octet = P! = qiqj - -6. qk qk, 3 3 l
Unitary Groups and SU(3)
152
The octet representation is a regular or an adjoint, representation of S U ( 3 ) because Pj transforms in t,he same way as the generators
Fj". Now we look at more general representations of SU(3). The general prescription for finding the basic tensors T:.::: for an irreducible representation of S U ( 3 ) is: 1. Construct tensors q?,:::. 2. Symmetrize among i l . - i, and j , . . j , indices. 3. Substract traces so that all contractions give zero, e.g. TZ
z
!2...!q 22"'tP
= 0, et,c,
The linearly independent, components of tensor T then supply an irreducible representation of SU(3) which is designated as ( p , q ) . The dimensionality of such a representation can be easily computed. First let us calculate the number of independent components for a symmetric tensor with p lower (or q upper) indices. We note that each index can take only the value 1, 2 or 3. Thus the number of independent components are the same as the number of ways of separating p identical objects with two identical partitions:
Thus a tensor which is symmetric in p lower indices and q upper indices has
B(Pd
=
( P + 2) (P + 1) ( 4 + 2) ( 4 + 1)
4
independent components. But the trace condition shows that a symmetric object with p - 1 lower indices and q - 1 upper indices vanishes. This gives B ( p - 1, q - 1) conditions. Hence a symmetric traceless tensor has dimensions
= (P
+ 1) ( 4 + 1 ) p(+
+ 1)
(5.66)
Irreducible Representations of SU(3)
153
Thus we have for example:
3 : Triplet
10 : Decuplet 27 : 27 plet
(2.2) Young’s Tableaux
By taking the direct product of basic representation 3 with itself, we can generate the representations of higher dimensions. These representations are however reducible. We now discuss a general method to decompose these reducible representations into irreducible representations. We have already discussed some simple examples. We represent the fundamental representation 3 by a box i.e. associate index i with a box.
0: 3 : qa. We note that the representation two 3’s viz.
(5.67)
3 is antisymmetric combination of
- E ~ J A’ ~j k ,
A 23. .
=
1
-&.. 2 23k
Tk.
(5.68)
This can be represented by a column of two boxes : 3 : Ti.
(5.69)
Unitary Groups and SU(3)
154
Since a tensor index takes on only three values i = 1 , 2 , 3 ,a column in Young's t,ableaii can have at most, three boxes (5.70) It, is completely antisymmetric and it, corresponds t,o the trivial singlet, representation. We note that, (5.71) Consider the ( p , q ) representation. It is a tensor whose components are (5.72) symmetric in lower and upper indices and traceless. We can lower the upper indices with E tensors, obtaining an object with p 2q lower indices:
+
h 11
*.. ...
k,
4
il
I
...
IZPJ
(5.75)
...
k,
21
Irreducible Representations of SU(3)
155
still correspond to the representation ( p , q ) . Comparison with the Young Tableau gives the following rule for preparing a tensor with the right symmetry properties to give a state in ( p , 4): First, symmetrize indices in each row of the tableau. Then antisymmetrize the indices in each column. If we have more general tableau with columns of more than two boxes, the rules for forming a tensor are the same as before. Assign an index to each box. Then symmetrize the indices in each row and finally antisymmetrize the indices in each column. Some of the common irreducible representations of S U ( 3 ) are shown in the Table 5. Decomposition of Product Representations We now consider the decomposition of the direct product of irreducible representations ( p , q ) and ( T , s) corresponding to tableaux A and B.
l a lI a lI a lI a lI a lI l b l b l b l
(5.76)
We now give a recipe for the decomposition of the direct product of ( p , q ) and ( T , s ) with the aid of Young Tableaux. Put a’s in the top row of B and b’s in the second row. Take boxes with a from B and add them to A, each in a different, column, to form new tableaux. Then, take the boxes with b and add them to form tableaux, again each box in a different, column, with one additional restriction given below. On reading the added symbols from right to left and from top to bottom, the number of a’s must be greater than or equal to that of b’s i.e. forget all tableaux which concave upwards or towards the lower left. This avoids double counting of tensors. The tableaux formed in this way correspond to irreducible representations in ( p ,q ) 8 ( T , s). We now give several examples to illustrate how this recipe works.
Unitary Groups and SU(3)
156
Table 5.5 'Irreducible r&resentations of S U ( 3 ) .
a
Tabular
Tensor
R
u
m
u
6
1x.4
Irreducible Representations of SU(3)
157
Examples (i)
0
m
3
6
Ed
(5.77)
9
(ii)
El
Ep
3
8
B
(5.78)
1
(iii)
3
8
.
3
@
1
(5.79) We discard the first and the third tableaux in Eq. (112) as they do not satisfy the constraint that number of a's greater than or equal to the number of b's as we go from right to left or top to bottom.
158
Unitary Groups and SU(3)
(5.80)
8@8=10$27@8$10$8@1 or ( 1 , 1 ) @ ( 1 , 1= ) (3,0)$(2,2)$(1,1)$(0,3)$(1,1)$(0,0). The slashed tableaux are discarded because they do not satisfy the const,raint a's 2 b's.
(4
15
8 @ 3 = 15@6$3
6
3
(5.81)
159
To summarize, an arbitrary irreducible representation of SU(3) is denoted by two integers, each positive or zero: ( p ,q). The corresponding irreducible tensor is denoted by ~ ~ It trans: . . $Tq. & . . . +iP. Each component of the tensor is an forms as eigenstate of 13 and Y and possibly of 12. If it is not an eigenstate of 12,such a state can be formed by a linear combination of states with components having the same 1 3 and Y . The basic states occurring in ( p , q ) can be completely labelled by three quantities 1, 1 3 , and Y , which form a complete commuting set within an irreducible representation. The values of I and Y that appear in ( p , q ) are given in Table 6. We note that highest state i.e. the one with I,, has
-
(5.83)
SU(N)
5.5
We now discuss Young’s tableaux for S U ( N ) . Again we assign an index to a box. Thus fundamental representation N ( d i , i = 1- - N ) is represented by a box:
-
u:N
(5.84)
~
~
Unitary Groups and SU(3)
160
Table 5.6 (P’ 4 ) .
lsospin I and hypercharge Y for the states in representation
I
Y
1 7
+ 3 -2
6
3
0
-1
1
-E-
1 1 0 -1 1 0 -1 -2
2
1To i
Numberof states
2 1 1 2 2 3+1=4 2
for the highest state I
1
1.3
I
I
2’ 3
2’
3
--+ 2
1 1
6
4 1
3 f 3 1 4 + 2 =6 2’ 2 -32 3+1=4 i, 0 3 1 -5 2 , ?L 2 1 3 3 1 4 + 2 =6 1 2’ 2 5+3+1=9 0 2’1’0 3 1 4+2=6 -1 2’ 2 -2 1 3
32 ’ 1
I
I
The tensor
~
i
~
~ is represented ~ . . . ~ by ~
a column of N boxes
(5.85)
It describes the singlet representation 1 of S U ( N ) . Now a completely antisymmetric tensor:
0, if any of the two indices are equal f l , if il . . i j ~ is an even (odd) permutation of 1 , 2 , N . The N dimensional representation ( N - 1) boxes.
N
1
E
~
~
~is ~
(5.86)
is described by a column of
: N.
(5.87)
Hence we see that for SU(2),2 and 2 are equivalent representation . Only for N 2 3, N and N and both will be represented by are distinct representations. We now discuss the decomposition of the product of repre-
.
.
.
~
~
Unitary Groups and SU(3)
162
sentation N by itself into irreducible representations of SU(N ) .
12 N ( N - 1). (5.88) Thus N 2 components decompose into two irreducible representations of dimensions and viz. N
N
=
iN(N+l)
(5.89)
where
(5.90a) (5.90b)
We can regard Stj as an N x N matrix, but since it is a symmetric matrix, it, has only + N = i N ( N + 1) independent, elements and this gives the dimension of symmetric representation Sij. Again if we regard A,j as N x N matrix, we can easily see that, N ~ - N it has 7 = f N ( N - 1) independent, elements and this gives the
9
163
SUP)
dimension of antisymmetric representation Aij.
N-1 Boxes
N Boxes
@ N = 1@ Adjoint
Boxes in the column
representation of dimension N 2 - 1. (5.91)
Thus
$i43. -- Ti j + 'z 6 2 , , $ $kT
(5.92a)
where 1 Sj 2 4, 4 k . (5.92b) N The adjoint representation has the same dimension as the number of generators of S U ( N ) . For example for S U ( 6 ) : 6 @ 6 = 1@ 35. We now give a general recipe to calculate the dimension of irreducible representations in the decomposition of the product of representation N by itself. To calculate the dimension of an array of boxes there is a recipe which involves calculation of ~ ~
Tj = @
4j -
Numerator: Insert N in each of the diagonal boxes starting from the top left hand corner of the tableaux. (5.93)
~
Unitary Groups and SU(3)
164
Along the diagonals immediately above and below insert N + 1 and N - 1 respectively. In the next diagonals insert N 2 and.so on. The numerator is equal to the product of all these numbers. For example for the tableaux
+
(5.94)
the numerator
=
N 2 ( N + 1) (N- 1) = N 2 ( N 2 - 1).
Denominator: The denominator is given by. the “product of hooks”. We associate each box with a value of the hook. . To find it, draw a line entering the row in which the box lies from the right. On entering the box, this line turns downwards through an angle of 90” and then proceeds along the column until it leaves the diagram. The value of hook associated with that box is then the total number of boxes that the line has passed through, including the box in question. The product of hooks is the denominator. We illustrate this by the following example. Consider the tableau (94). The hooks associated with each box are shown in Eq. (95).
3
2
2
1
(5.95)
We see that the denominator = 3 x 2 x 2 x 1 = 12. Hence the tableau (94) corresponds to an irreducible representation of dimen-
sion N 2 ( N 2 - 1)/12. Let us now consider some more examples:
N e
N 8
-
N
(5.96)
To avoid double counting, we discard the slashed tableaux. Thus we can write (5.97) Note further that
qijp + q k i ] j + q k ] i = 0.
(5.98)
In order to find the dimension of these representations, we note that
N
N
N
@
(N11 N(N+ I ) ( N + l ) 6
N(N+ 1)(N-1) 3
N(N+ 1)(N-l) 3
N(N- 1)(N-2)
6
(5.99)
Unitary Groups a n d SU(3)
166
For example for SU(6) : 6 @ 6 @ 6
= 56 @
70 @ 70 @ 20.
(2)
B
N(N+I)(N-I)(N-2)
12
4x2~1
N@J-l)@J-2)@J-3)
4x3x2x1 (5.100)
Hence we have
U N(N-1) N(N-I) 2 2
N2(N2- 1) NrN2- 1)(N-2) N(N-1)("-2)W-3) 12 8 24 (5.101) For example for SU(3) : 3 @ 3 = 6 @ 3 and for SU(5) : 10
@I0
=
50 @
45
@35.
Applications of Flavor SU(3)
5.6
167
Applications of Flavor SU(3)
1. SU(3) Invariant BBP Couplings If OA is an octet operator, the matrix elements of this operator between the states 18,B ) and 18, C) can be written as (8,C PA( 8, B ) = i
~ A E CF
+ ~ A E CD .
(5.102)
That there are two independent couplings follow from the fact that as noted previously 888 contains 8 twice. In particular if OA is pseudoscalar meson octet operator PA,and 18, B ) , 18,C) are octet of baryon states, the BBP couplings can be written as gABC
= 29 [i fABC
f + dAEC dl
a
(5.103)
For example
f +d3--d] 4 + i 5 4 - i5
JZJZ = g (f
+ d ) = -gnonn
We normalize grow = g, so that, f
9r-W
a*
=-
(5.104)
+ d = 1. Then (3f
9 + d ) = --fi (1 + 2 f ) .
(5.105)
In this way we can calculate all the relevant couplings:
168
Unitary Groups and SU(3)
Experiment,ally
2. VPP Coupling Here we take OA = VA, the vector meson octet and 18,B) and I8,C) are octet of pseudoscalar meson st,ates. Now under charge
conjugation C:
VA
IW)
--t --f
-VAVA,(no summation over A ) 7lf3 I 8 J ) >
(5.108)
Hence the invariance under charge conjugation gives "YF
i
fABC
4- Yo ~ A B C ]
(8,
c lv~l8,B )
---t
-?A
-
-VA
c lv~l8, B ) VB V C [ Y F i f A B C + V B VC
(8,
dABC].
(5.110) But [cf. Eq. (30)]
(5.111) Therefore. we have 7~ = -70
or yo = O .
Hence V P P has only F-type coupling. Thus (8, C IVAJ8, B ) =
~ A B C27,
(5.112)
Applications of Flavor SU(3)
where we have put
169
For example & = po:
YF = 27.
1-22
l + i 21-22
2 --3f
JZJZ
27
= 27.
(5.113)
Thus ypSS = 27. It is straightforward to calculate the V P P coupling for other members of the octet,, which are given below:
The decay width of decay V
-+
P P is given by (5.115)
Hence we have
r (p
-+
This gives
m)= M
rtot(K*+
--t
5 s(G) = 149.1 4 7 ~ 3 rn;
2.9 MeV.
(5.116)
0.74. Now
K ~ = )
r (K*+
--t
~ O K +
>+ r (K*+- + ~ + K O )
This gives rtot(K*' 4 KT) M 44.5 MeV to be compared with the experimental value 49.8 f 0.8 MeV. In broken S U ( 3 ) , w8 can mix with the singlet wl, so that the physical particles w and (b are linear combinations of w8 and w1:
4
( t)
= w8cosB - wlsinB
case -case sine)
= (sine
(
$L
(5.118)
Unitary Groups and SU(3)
170
We now show that w1 + P P is forbidden by charge conjugation invariance. The invariant coupling in this case is w1,
P;
a, p,j
which changes sign under charge conjugation. Hence
r (4 + K + K - ) = C O S ~e r (u8-+ K K ) Therefore,
and
where we have used cos26’ = 2/3 [cf. Eq. (153)]. This gives r ($ --t K f K - ) = 1.95 MeV to be compared with the experimental value 2.1 MeV.
5.7
Mass Splitting in Flavor SU(3)
In exact, SU(3), the particles belonging to an irreducible representation of S U ( 3 ) must have the same mass. But, we note that, all members of a supermiiltiplet, do not, have the same mass. This means that SU(3) is not, an exact symmetxy of strong int,eractions, but is a broken symmetry of these interactions. This means that, the interaction Hamiltonian consists of two parts viz.
H
= Ho
+ Hl,
(5.12la)
where
(5.121b) (5.121~)
Mass Splitting in Flavor SU(3)
171
i.e. Ho is SU(3) invariant, but we take H I such that
[I, Hl] = 0,
H1
breaks the S U ( 3 ) symmetry. If
[Y,Hl] = 0,
(5.122)
H1 still preserves the isospin symmetry and hypercharge is conserved in its presence. The first of Eqs. (122) holds only in the absence of electromagnetic interaction. In order that S U ( 3 ) to be meaningful, HI must be at least an order of magnitude weaker than
HO
*
The simplest general form of H1 in S U ( 3 ) which satisfies Eqs. (121c) and (122) is
(5.123) To get H I from the quark model, the mass Hamiltonian for quarks is given by H, = mu U u + m d d d m, 3 s,
+
where mu,m d and m, are masses of u-quark, d-quark and s-quark respectively. In the exact S U ( 3 ) limit, mu = md = m,. If SU(3) is broken but isospin symmetry S U ( 2 ) is still exact, then mu = md # m,. Now we can write tiu-dd m ( t i u + d d ) + m , ~ s + ( m , -md) 2 2771 + m , (u u d d 3 s) 3 m -m, 1
Hq =
+
+
+
(tiu+dd-2~3)
(5.124) where
Unitary Groups and SU(3)
172
Hence we can write (5.126) This also shows that, SU(3) symmetry breaking term transforms as A8 under SU(3). It was shown by Okubo that for any irreducible representation ( p , q ) of SU(3), the matrix elements of tensor T i are given by
’:[
( ( p ,q ) I , Y IT:/ ( p ,q ) I , Y ) = a+bY+c - - I ( I
1
+ 1)
,
(5.127)
where a , b, c are independent of quantum numbers I and Y but, in general depend on ( p , 4 ) . Thus we can write the mass formula for particles in a multiplet of SU(3) as m = mo
+ Am = a + by + c ’:[- - I ( I + I)] .
(5.128)
Let 11s apply this formula to baryon octet. Then from Eq. (128), we get, 3m~+mc mN+mz (5.129)
4
2
whereas for pseudoscalar meson octet, we get (5.130) In Eq. (130), we have used squared masses, as in the Lagrangian for bosons, the square of boson masses appear. Equations (129) and (130) are well known Gell-Mann-Okubo mass formulae. For the decuplet
Y
I=-+l, 2
(5.131)
173
Mass Splitting in Flavor SU(3)
and Eq. (128) reduces to
m = a'
+ b'Y
(5.132)
and we obtain the equal spacing rule for t,he decuplet:
mn - m B .
= ms. - mc. = mc* - ma.
(5.133)
The mass relations (129) and (133) are well satisfied experimentally and are regarded as a great success of SU(3). Similarly for vector
(5.134) Since due to mixing between w8 and the singlet w1, the physical particles are 4 and w , the formula (134) is not directly applicable. We will come to this formula later. Similar remarks are applicable to the mass formula (130). For octet and decuplet representations of SU(3), one can easily derive the mass formula as follows. We note from Eq. (126) that
(5.135) where
q.is an octet operator viz. 0" = 4%q . - -6, ' 2 q k qk 3 3 3
(5.136)
and T8
A8
= -.
2 Hence we see that HI transforms under SU(3) as
(5.137)
Unitary Groups and SU(3)
174
Thus to first order in X , the mass splitting for the state ) A ) of an SU(3) multiplet is given by
Am
X (AJHIJA) = X (A/O;/A). =
(5.139)
Let, us apply it to baryon octet:
Hence we have
This gives the Gell-Mann-Okubo mass formula (129) for baryons. For the decouplet, we have X 0; = x
T"j3Zj3.
(5.142)
This is the only possibility as zjk is a completely symmetric tensor.
Mass Splitting in Flavor SU(3)
175
This gives ma = mo m p = mo+2X msb = mo+4X mn- = mo+6X,
(5.144)
and hence we have the mass relation (133) for the decuplet. For octet of vector mesons
x
(-5
WE)].
(5.145)
Hence we have mz = mi mK. 2 = mi+XOD 2 4 rnwe = r n i + - X O o . 3
(5.146)
This gives the octet formula (134) for vector mesons. Now W E and mix, when SU(3) is broken, the mass matrix in W E and 01 basis
w1
can be written
(5.147) Using Eq. (118), we can diagonalize it:
UTM2U = where
u=
(
(
m; 0
O2 mu
cos0 -sin0 sin8 cos8
),
(5.148)
(5.149)
Unitary Groups and SU(3)
176
This gives
m$+m: = rn;+m;
(5.150a)
mb - mi = (cos2o - sin2O) (mi - m:> 2
+4 sin o cos o mPs,
(5.150b)
2 m5s mi - m:
tan28 =
(5.151a)
3 m: - 4 m:, +m; tan2@ = m: - mi 2 . (5.151b) 4 mK*- m; - 3 mz ma - m; Now using mK* = 892 MeV, mp = 770 MeV, m, = 783 MeV and mb = 1020 MeV, we have ma E mw8= 930 MeV, ml = m,l = 880 MeV and tan0 M 0.84, 0 M 40'. (5.152)
It is tempting to take 1 tan0 x - M 0.71,
1/2
For this case sin0 x 4'cos0 M
I4
(5.153)
$ and
Jz IW1) = 6 I%> + 6 1
=
0 M 35.3'.
-
1 ";2d
d)
.
(5.15413)
Hence we have 2 m,2 = mp
(5.155a)
and from Eqs. (151b) and (153) m4 2 - m,2 = 2 (m;* - m:)
.
(5.155b)
177
Mass Splitting in Flavor SU(3)
K
5 I
Figure 6 a)$ --t K K decay allowed by OZI rule, b): suppressed by OZI rule
4
decay
Equation (153) gives the “ideal mixing”. With this mixing of strange quarks only. Experimentally it, is observed that 4 t pn or 3n is very much suppressed as compared with 4 t K K . Note that p~ or 37r do not, contain any strange quark. The suppression of 4 decay into non-strange particles is explained by the so-called Okubo-Zweig-Iizuka rule (OZI rule) : “The decays which correspond to disconnected quark diagrams are forbidden”. Thus the decay in Fig. 6a is allowed but the decay in Fig. 6b is forbidden. There is no theoretical basis for the OZI rule. No strong interaction selection rule forbids the
4 is made up of s d i.e.
Unitary Groups and SU(3)
178
decay of q5 -+ p r or 37r. But experimentally this rule seems to be well satisfied. The small decay width for q5 -+ p7r (37r) can be explained by some deviation from the “ideal mixing” which allows small admixture of non-strange quarks in q5. 5.8
Problems
1. Show that for a vector operator Oi(2 = 1 , 2 ) under SU(2)
Given
( a ,3/2, -1/2
P , 1, -1)
1 0 1 1
= F,
find ( a ,3/2, -3/2 1 0 2 1 P , 1, - 1 ) . The states are labelled as la, I , I s ) . ff2. Suppose that ( 7 ~ ) : .= ( 7 ~ and ) ~ ( ~0 ~ = ) (~a ~ )are , ~Pauli matrices in two different two dimensional spaces. In the four dimensional product space, define the basis vectors
Jp= 1) =
12 =
Ip = 3) =
12
1) la = 1 ) , lp = 2) = 12 = 1) la = 2) = 2) lo! = 1 ) , lp = 4) = ( 2 = 2) la = 2 ) .
Define
TAB= TA 8 a ~ (TAB): ; = ( 7 ~ ) ;(OBI; , p , ~ = l , * * * , 4 ,A,B=l,2,3. Evaluate T21 = (7-2 @I a l ) , as a 4 x 4 matrix.
3. A second ranked mixed tensor Ti transforms as the unitary transformation
4; show that
= a:
$j,
[F;, q k ] = 6;T3”- 6; q .
4: &, under
Problems
179
4. (a). Using the following relation for SU(3):
show that
(b). Using the relation
and Eq. (60b) of the text, show that
5. From the group property v-’(b)U ( a ) U ( b ) = U(b-lab), derive the commutation relation for the generators of the unitary group U ( N ) :
6. Show that XI, A2 and A3 generate an S U ( 2 ) subalgebra of SU(3). Show that the representations generated by the remaining A’s or their linear combinations transform as doublets and singlet representations of SU( 2).
generate an SU(2) subalgebra of 7 . Show that X2, A5 and SU(3). Show that the representation generated by the linear combinations of remaining A’s transform as 5-dimensional representation of SU(2). Hint: A5 2 x 2 act as raising and lowering operators.
Unitary Groups and SU(3)
180
8. Find the matrix generators XA ( A = 1.. . 15) for the group SU(4). 9. The following assignments for 3 quarks are given instead of usual ones:
u‘
B 1
d’ s’
1 1
S -2 -2 -3
I
13
1/2 1/2 0
1/2 -l/2 0
Find the charge Q and hypercharge Y for each quark in this case. Mesons can be constructed as ij’q as before. If baryons are constructed as q’q’q’, can the above assignment, of quarks work? If not) discuss the difficulties encountered.
10. Find the U-spin eigenstates for the baryon octet, and decuplet. Plot them on Q versus U3 plot.
11. As far as S U ( 3 ) is concerned, magnetic moment operator transforms as T: which is singlet under U-spin. Using this fact and the U-spin multiplets found above, show that the baryon octet magnetic moments are related as follows:
pLCo-Ao
d3
--(pA-pc).
2
12. In S U ( 3 ) , find 1088,
10810,
8@3.
13. In S U ( 5 ) , show that
5 @ 5 = 24 @ 1, 5 @ 10 = 5 $45,
10 @ 10 = 5 @ 5 0 @ Z 10 8 10 = 1 @ 24 @ 75.
Problems
181
14. Consider the representation 6 of SU(3). Writ,e down the particle content of this representation in terms of quarks. If s U ( 3) breaking Hamiltonian HI transforms as o3or T8,write down the mass formula for these particles.
15. Draw the weight diagrams for the 15 plet and 27 plet representations of SU(3) 16. Consider the 0- nonet. Experimental masses are m, = 137 MeV, mV = 549 MeV,
m K = 496
MeV, = 958 MeV.
mVj
From the octet, mass formula, find mq8.Compare it2with m?. Assuming that discrepancy between the two values is ent,irely due to 71 -"q8 mixing in broken S U ( 3 ) , so that 17') = COSO
+sin6
1778)
Jq)= - sine 17') + c o d lq8) ,
,
find from the experimental masses and m,, rnV, and the mixing angle 8. If we write 17) = ~
0 I q~n s } 4-sin4
,
1~s)
17') = sin 4
, the
values of
I q n s ) +COS
4 17s) ,
where
show that
4 = tan-' fi + O.
17. You are given an octet, operator
+
0 = cos O O~+ia sin O
04+i5,
determine the S U ( 3 ) matrix elements for the transitions:
n t p
,
zo ,
C--+n,
E0+A , in terms of F , D and 6. zu
C-th
-=O+c
, ,
co- ) p z- c+ t
Unitary Groups and SU(3)
182
18. Write down the D B P couplings in the SU(3) limit for the process D-Bp where
D : Bayron decuplet, J p = 3/2+ B : Baryon octet, J p = 1/2+ P : Meson octet J p = 0-. Hence show that for the energetically allowed decays, they are in the following ratios: =*'=*o A++ C*+ c* u 4 p7r+ ' -+AT . 4 C T * -+=-7r+ .* --$ s * 0 7 r o Y
- & : & i :1
fi
-1
Bibliography
5.9
183
Bibliography 1. M. Gell-Mann and Y. Ne’eman, The eightfold way, Benjamin, New York (1964). 2. S. Okubo, Lectures on Unitary Symmetry, (unpublished Univ. of Rochester Rep.). For SU(3), we have drawn heavily on this reference. 3. P. Carruthers, Introduction to unitary symmetry, Interscience, New York (1966). 4. D. B. Lichtenberg, Unitary symmetry and elementary particles (2nd edition), Academic Press, New York (1978). 5. F. E. Close, An introduction to quarks and partons, Academic Press, New York (1979). 6. R. Slansky, Group theory for unified model building, Physics Report 79c, l(1981). 7. H. Georgi, Lie algebra in particle physics, Benjamin Cummings, Reading Massachusetts (1982).
Chapter 6
SU(6) AND QUARK MODEL 6.1
SU(6)
Quarks have spin 1/2. The well known baryons with spin 1/2 and spin 3/2 are in the octet and decuplet representations of flavor SU(3). We note that within each representation the mass splitting between adjacent members is of the same order. For example mA - m N
M
mc. - ma
M
170 MeV, 153 MeV,
mz - mc = 125 MeV; mn - m g . = 142 MeV.
It is tempting to put these two representations in an irreducible representation of the group higher than SU(3). But octet and decuplet representations have different spins. This means that the proposed group cannot commute with angular momentum (spin). The proposed group must contain SU(3) x SUo(2) as its subgroup. This might cause some trouble, since we are combining an internal symmetry with a space-time symmetry. It does cause trouble but this does not show up until one tries to make the theory relativistic. We note that spin 3/2 baryon decuplet has (10 x 4) states and spin 1/2 baryon octet has (8 x 2) states. Thus, we look for an irreducible representation with 56 dimensions. Such a representation occurs in the decomposition of the product of representation 6 of SU(6) by itself viz. 6 @ 6636 = 56 @ 7 0 @70@ 20.
(6.1) The representation 56 is completely symmetric irreducible representation of SU(6). The six quark states (in this section we will 185
SU(6) and Quark Model
186
not write 1 ) explicitly) (u t u J, d d J s T s J) can be put in the fundamental representation 6 . We denote such a state as \ki, : Q = 1,2; i = 1 , 2 , 3 . In matrix notation we write
Now SU(3), SU(2) and SU(3) xSU(2) are subgroups of SU(6). The representation 6 splits under these subgroups as shown in table below:
SU(3) x SU(2) Thus, we see that SU(6) has 35 generators. Hence the adjoint representation of SU(6) has dimension 35 and is given in the following decomposition: 6 @ S = 35 @ 1.
(6.3)
The representations 56 and 35 split under the subgroup SU(3) xSU(2) as follows:
187
35
:
[(3,2) 0 (3, 2)] -
(8, 3)@(I7 3) @ (8, 1) octet of nonet pseudoscalar of vector mesons mesons
@
(1, 1)
singlet pseudoscalar meson (6.5)
SU(6) Wave Function for Mesons The mesons are composite of 477. The lowest lying mesons have
(@),=,
and
P = (-1) (-1) 0 = -1.
The spin wave functions are given by:
1
Spin singlet state : XA =
Jz r L sr) -
(6.6a)
The spin singlet state is antisymmetric, it gives J p = OW, whereas the spin triplet states are symmetric and gives J p = 1-. Thus we can write for 0- and 1- mesons the state functions as given in Tables 1 and 2 respectively. Lowest lying baryons are made up of three quarks: ( q q q ) L = o , P = (-1)0(1)3 = 1. Here we have to combine three spin 1/2’s. In this case we have the following decomposition:
SU(6) and Quark Model
188
‘ Particle 7r+
SU(6) State L (uTJl - ulJT’) r=
4
2
Completely Symmetric Spin3/2
Mixed Symmetry Spinl/2
2 (6.7) Mixed Symmetry Spinl/2
It is convenient to combine first two spin 1/2’s. For this case we have S = 0 and spin wave function X A (Eq. 6a) and S = 1 and spin wave functions xs (Eq. 6b). We now combine spin 0 with spin 1/2 and we get the spin S = 1/2 and the following wave function XMA:
We now combine spin 1 with the remaining spin 1/2. For this case we get S = 3/2 and S = 1/2. The spin wave functions for this case are given in Table 3. In this table, the numerical coefficients are Clebsch-Gordon Coefficients in combining spin 1 and spin 1/2. The state function for the completely symmetric represen-
189
Table 6.2 Vector meson states: ( q ~ ) L = o , J p = 1-.
Particle SU(6) State P+ PO
P-
K*+ K *O K'-*0 K (J
4 J,_= 1 P+ PO P-
K'+ K *O
K*-*0
K
W
4
ufdf
SU(6) and Quark Model
190
Table 6.3 Spin wave functions for S = 3/2 and the combination of spin 1 and spin 1/2. S, = 3/2 = 1/2 S,= -1/2
s,
S = 1/2 resulting in
S, = -312
312
xs
Symmetric:
ITIT)
7
I
in 1 and 2)
I
tation 56 of SU(6) can be written: @S X S f
1 - [ @ M S XMS -k @ M A
Jz
XMA]
1
(6.9)
where [cf. Eqs. (5.96), (5.91) and (5.92)]
@s= (T,jk)
z; lo),
(6.10a)
q.10).
(6. lob) The spin state functions XS, XMS and X M A are given in Table 3 and Eq. (8). Using Tables 5.3 and 3, we can write the state function as xs for the decuplet. For example, @MA
=
@MS = -
(A', S,= 1/2)
Similarly we can calculate all the other states. These states for
S, = 3/2 are given in Table 4. For baryon octet 1/2' states, we use Tables 5.2 and 3 and Eq. (8). We explicitly calculate the state ( p , S, = 1/2) . It is given by
-
2-{ ( - 1 )
Jzz
[(ud + d u>u - 2u u d ]
SU(6) and Quark Model
192
In a similar manner we can calculate the rest of the states. They are given in Table 5. Finally we give the state functions for the representations 70, 70 and 20. They are as follows: Representation 70 : MS @s X M S : (10, 2) : 20 @MS x s : ( 8 , 4) : 32 1 3 (-@MS X M S @ M A X M A ) : (8, 2) : 16 @ A = IAy)[cf. Eq. (5.95)]. @ A X M A : (1, 2) : 2, Representation 70 : MA @sX M A : (10,2) : 20 @MA XS (814) : 32 $ ( @ M s X M A + @MA X M S ) : ( 8 , 2) : 16 @ A X M A : (1, 2) : 2 Representation 20 @ A X S : (1, 4) : 4 1 3 ( @ M S X M A - @ M A X M S ) : ( 8 , 2) : 16 We will not give the detailed identification for these states.
+
6.2
Magnetic Moments of Baryons
Magnetic moment, operator is given by
ii
= SPO J
/ ti.
(6.15)
We define the magnetic moment p of a particle of mass m: P = SPOJ,
(6.16)
Magnetic Moments of Baryons
Table 6.5
Particle State
193
SU(6) states for the octet of bayrons J p = 1/2+.
SU(6) State functions
-2dtUJd - 2d'dtuJ -2~ldTdr dTdW
+
Remarks
Change uc-td and over all sign in p
Change d-isinp
SU(6) and Quark Model
194
Particle State
IC-,
I-O,
s, = 1 / 2 )
SU(6) State functions
2 6 8
s, = 1 / 2 ) 1 rn
Remarks
Change u 3 d and over all sign in [-C']
Change d 4 s and over all sign in n
Change u + -d in 30
195
Magnetic Moments of Baryons
where po = eh
/
(6.17)
2mc
and J is the angular momentum viz. the eigenvalue of l)h2. For electron, J = 1/2, g = -2, i.e., p e = -eh
/
2mec.
J2 is J ( J + (6.18)
For a spin 112 particle, J = 1/2fw. Thus for a quark, the magnetic moment operator is given by
(6.19) where (6.20) is the magnetic moment of the quark. The magnetic moment operator for a baryon of J p = 1/2+ in the quark model is given by (6.21) We need to calculate the expectation value of
@Bz
viz.
We now explicitly calculate the magnetic moment of the proton. For the proton
SU(6) and Quark Model
196
Using the proton state Ip,az = 1) as given in Eq. (13), we have (we will not write (T, = 1 explicitly in the state)
1
GPZ
IP) =
\/Is
Hence
(6.25) Similarly using Table 5, we can calculate the magnetic moments for the rest of the baryons in the octet. However, we can use simplified state functions to calculate the magnetic moments. In this calculation the order in which quarks appear is important. For the proton, we write the state function
Magnetic Moments of Baryons
197
Hence
(6.28)
Therefore,
PP -
1
(@PAp
= g [8PU
+ (1+ 1 - 4) P d ]
4 1 - ~ P -u - Pd* 3 For [Ao)>,the simplified state function is given by 1/2 1 lAo) = - Iu d s> X M A = - Iu d s) - I(T1- 1t) T)
&
(6.29)
Therefore, P A = ( F A ~ ) A=
51 [O + 2 pS]= ps.
For ICO), the simplified state function is
(6.32)
SU(6) and Quark Model
198
Therefore,
(6.35) From Eqs. (31) and (33), we get
1 - [Pu - Pd] = PAO-EO.
(6.36)
The magnetic moments for the rest of the baryons in the octet can be written from Eq. (24) as follows:
In order to compare these magnetic moments with their experimental values, we introduce the following quantities: eh
Po = -
2mc '
We can write Po = PN
-
m=
mufmd
2
(z) ,
(6.42)
(6.43a)
199
Magnetic Moments of Baryons
where (6.43b) Here P N is the nucleon magneton. Thus we can write the magnetic moments of u, d and s quarks in terms of P N :
2 -(-) mu
Pu = 3
mp
PN
(6.44a) (6.44b) (6.44~)
We will now assume isospin symmetry i.e. will take mu = md = m.We see that we have two unknown numbers 5Fi and m,. These numbers we fix from the experimental values of p p and p ~ . From Eqs. (24), (31) and (43), we obtain
'-= m
p - mP
= 2.793 p ~ .
(6.45) (6.46)
On the right hand side of Eqs. (44) and (45), we have put their experimental values. Fkom Eqs. (44) and (45), we get m = mu = md M 336 MeV ma x 510 MeV.
(6.47a) (6.47b)
It is interesting to compare these values with those obtained from the naive quark model. Now proton is made up of uud quarks and A is made up of uds quarks: 3 TE = mp, 2Wi+ma=m~,
ma=
-
m x 313 MeV
3 m~ 2mp = 490 MeV. 3
(6.48a) (6.48b)
200
SU(6) and Quark Model
The masses of u, d and s quarks given in Eq. (46) or (47) are called the constituent quark masses. These are effective masses of the quarks confined in a hadron. The constituent quark masses are quite different from those appearing in the Hamiltonian or the Lagrangian. These masses are called current quark masses. Using Eq. (46) and (43), we get (6.49a) (6.49b) (6.49~) Using Eqs. (48) the predictions of quark model for the baryon magnetic moments as given in Eqs. (24), (31), (34), (35) and (36)(40) are tabulated in Table 6 along with their experimental values. If we put mu = md = m,, in Eqs. (24), (31), (34), (35) and (36)(40), we get the SU(6) predictions
We conclude this section by the following observations: 1) The quark model is simpler than SU(6). 2) It is more predictive than SU(6). It gives information about the scale of magnetic moments. 3) It gives good account of some corrections to SU(6) relations. From Table 6, we see the agreement between quark model values of baryon magnetic moments and their experimental values is not bad. 6.3
Radiative Decays of Vector Mesons
For a quark and antiquark system, the Hamiltonian is given by
201
Radiative Decays of Vector Mesons
Table 6.6 Magnetic moments of baryons: Quark model predictions
To introduce electromagnetic interaction, we make the gauge invariant replacement
6
+
@-eQA(r,t),
(6.52)
where A (r, t) is the electromagnetic field, e& is the electric charge of the quark and 3 = -ZV. From Eqs. (50) and (51))we get
(6.53) Using the identities (0
- 6 )( u *A) + (U* A )( 0 -6 ) = 6 6.A
and the gauge condition
*
A + A.8 +
= As@- V .A
20.
(-ZV x A ) (6.54a) (6.54b)
SU(6) and Quark Model
202
we write Eq. (52):
H = HO+ Hint,
where
2
(6.55)
i
(6.56)
H. - ,nt
- e
-[2A (ri,t ) - Ga Qi
2mi
+ iai
(-iVi x A (ri,t ) ) ].
(6.57) In Eq. (56), we have neglected the second order term e2. Now
where E” is the polarization vector, ax! (k’)and a:, (k’)are the annihilation and creation operators for the photon respectively. They satisfy the commutation relation [ax (k) a:,
(k’)]= 6xxt 6(k- k’).
(6.59)
We now consider the emission of a photon viz. the process
a+b+y.
(6.60)
We note that (6.61a)
(6.61b) ( b y ( a!, (k‘) = ( b I ax (k) a!, (k’) = ( b 1 [6x,, S (k- k‘)-
(k’)ax (k)]. (6.61~)
It is clear from Eq. (60) that only second half of Eq. (57) contributes and the matrix elements for the process (59) are given
203
Radiative Decays of Vector Mesons
by
x [2€*X
*
& - iaa (k x *
E”)]
) .1
eiWt,
(6.62)
where we have used
-V x A 0:(-i)2 k x eX*.
(6.63)
In &. (61), the term with 2 eA* & gives the electric transition and the term ui (k x E * ~ ’ )gives the magnetic transition. We now make the dipole approximation so that in the expansion e-ik.ri (6.64) - 1 - i k .ri ,
+
we retain only the first term. Then (6.65) Now
iia
i- = [ri, Ho] ma
+ O(e).
(6.66)
We go to the center of mass (c.m.) frame and introduce r = rl -r2 m ~ r ‘17~2212
R =
+
ml +m2 1 1 - = -+-, 1 P ml m2 In the c.m. frame R = 0, so that
[R, H ] = 0.
(6.67a) (6.67b) (6.67~)
(6.68)
Therefore, we have from Eqs. (64)-(66): (6.69)
SU(6) and Quark Model
204
where we have used the fact that) . 1 with eigenvalues E, and Eb:
and Ib) are eigenstates of HO
Hob) = GzJa), HO (b) = Eb) . 1 1
E, -Eb
=
(6.70a) (6.70b) (6.70~)
W.
We shall make use of Eq. (68) later. Here we consider the magnetic transition in dipole approximation i.e. allowed M1 transition. For M1 transition we get from Eq. (61)
We consider the decays of the form
V-Pfy 3s1 4 9 0
+’
+y.
(6.72)
For the transition 3S1 So , A L = 0 and there is no change in parity. Therefore, it is M l transition and the Hamiltonian given in Eq. (70) is relevant for the decay (71). Now we can write 0
1
(k x
=
E”)
gz
(k x
+&
+ Jz S+
E”’)
s-
(kx c ” )
(k x
c”)
(6.73a)
+’
where
1
SS
(k x
=2 (ax
E”’),
1
+ iay)
s- = 2 (a, - 20,)
a [(k x
E”)
)
=1
f i (k x
(6.73b)
)
E’*)~]
.
(6.73~)
If we take the matrix elements between V ( S z= 0) and P(S, = 0), we need to consider oiz (k x E ” ) i.e. we have to calculate the matrix elements of the operator
Cz=C (Qi/2mi)n i z a
(6.74a)
205
Radiative Decays of Vector Mesons
between the states
IV, S, = 0)
and
IP) .
(6.74b)
Using the state functions given in Tables 1 and 2, we can easily calculate the matrix elements (&) . We explicitly calculate ( j i z ) for the transition w o T O . It is convenient to write --f
,'wI
Sz = O
1 )=3 1u T ~ +
da)
&.
(6.75)
Then iiz
Iwo,
s z = 0) (6.76)
(6.77) Hence we get (6.78) Similarly we can calculate &) for other members of the octet. They are given in Table 7. We now calculate the decay rate for V + Py. According to Fermi Golden Rule, the decay rate is given by
r = 2T 1 ( P I~if;;z1Iv)I p ( E ). If we consider the decay of the vector meson at rest, then
(6.79)
SU(6) and Quark Model
206
Table 6.7 The matrix elements (PI & (V, S, = 0 ) for M1 transition for the decay V -+ P y.
+
and
-
Vw2 Ep -dR, (2n)3 mv
[mv = E
p
+w] .
(6.80)
Now (PI Hzl ( V )is given in Eq. (70) with a = V and b = P. In order to calculate I', we have to average over the initial spins of vector meson V and sum over the final spins of the photon, The vector meson has three spin orientations S, = +1,0, -1. Instead of calculating ( H Z ' ) for S, = f 1 , O and then taking the average, it
0
is more convenient to calculate Hi%' for S, = 0 and forget about the spin average. Thus from Eqs. (70) and (73), we get
(6.81)
207
Radiative Decays of Vector Mesons
From now on we will not write S, = O explicitly in IV). We note : the following properties of the polarization vector ex
*
&A'
= Sxx,
k . e x = 0,
X = 1,2
(6.82) Using Eq. (81)) we have
I(k x X=l, 2
&*x> l2
= k2 (1
- cos20)
(6.83)
and
87r I d 0 lc2 (1 - cos20) = - k 2 . 3 Hence from Eqs. (78)-(80) and (83)) we get
(6.84)
(6.85) For the decay
P+V+y,
(6.86)
we only sum over the spin of vector meson and do not take the average. Hence for this decay, we have
r p+v+y)=4a
I ( V I ~ ~2 ~IC 3Ev I-.P ) I
(6.87)
mP
We note that a relativistic treatment of the phase space gives the expressions (84) and (86) without the factor Ep/mV and Ev/mp respectively. Thus we can write Eq. (84):
(6.88) where R is the overlap integral. It is of order 1, but it may differ from 1, if we take into account the distortion of wave function due
SU(6) and Quark Model
208
Table 6.8 Quark model prediction for V-+ P
I
Decay
I'
I
+ y with R = 0.735. r Experimental (in keV) f50.7 716'6 -49.8 50.3 f 5.3 114.5 f 11.8
to symmetry breaking introduced by the quark mam differences. R may vary from process to process. We assume that, this variation is not, large. Then we can fix 0 by using one decay, which we take p* 4 T* y. Using Eq. (87), Table 7, mu = rnd = 336 MeV and k = 372 MeV, we get
+
r (p*
-+ T*
+ 7 ) = (123 keV) 52'
(6.89)
+ y) = (67.1 f.8.8) keV.
(6.90)
But,
rexp(p*
4
T*
F'rom Eqs. (88) and (89), we get,
f2 z 0.735.
(6.91)
Using this value of R and rn,/m, = 0.66, we can compare the predictions of quark model using Table 7 and Eq. (87) with their experimental values. This is given in Table 8. We notice from Table 8 that agreement between the predictions and experiment, is only fair. This is underst>andablesince the relativistic corrections become important for hadrons involving light quarks [see, for instance Ref. 6 in the bibliography].
209
Problems
6.4
Problems
1. In quark model, using SU(6) wave functions, show that the Fermi matrix element for n + p transition: (P,
s z = ; ( c f / n ,s
- - 1) = l a
x-2
Q
Find the Gamow - Teller matrix element
2. Show that the transition moment between As and p is given by
3. Calculate the decay rates for the following decays in quark model:
4+77+Y 77'
--$ --f
PO + Y wo+y
and compare them with their experimental values (54.9 f 6 . 5 ) keV, (72 f 1 3 ) keV, (72 f 1 3 ) keV and (6.5 zkl.0) keV respectively. [You may take 778 - 771 mixing angle as 8 = -11O.I 4. Consider M1 transition decay Co -+ A'
+ y.
Calculate its decay rate in the non-relativistic quark model and compare it with its experimental value T
= (5.8 f 1.3) x
sec.
SU(6) and Quark Model
210
Hint: M1 transition operator is
Bibliography
6.5
211
Bibliography 1. M. Gell-Mann and Y. Ne’eman, The eightfold way, Benjamin, New York (1964). 2. J. J. J. Kokkedee, The quark model, Benjamin, New York (1969). 3. 0. W. Greenberg, Ann. Rev. Nucl. Part. Science 28, 327 (1978). 4. F. E. Close, An introduction to quarks and partons, Academic Press, New York (1979). 5. Particle Data Group, European Physical Journal C3, 1 (1998). 6. S. Godfrey, N. Isgur, Phys. Rev. D32, 189 (1985); V. 0. Galkin and R. N. Faustauve, Sov. J. Nucl. Phys. 44, 1023 (1986) and Proceedings of the International Seminar ”QUARKS’ 88”, USSR May (1988) p. 264.
Chapter 7 COLOR, GAUGE PRINCIPLE AND QUANTUM CHROMODYNAMICS 7.1 Evidence for Color As we have discussed in the introduction in order that 3 quark wave function of lowest lying baryons satisfy the Pauli principle, each quark flavor carries three color charges, red ( T ) , yellow (y) and blue ( b ) i.e. a = r, y, b. Qa Leptons do not carry color and that is the reason why they do not experience strong interactions. Thus each quark belongs to a triplet representation of color SU(3), which we write as SUc(3). Now SU(3) has the remarkable property that 3633633 = 10@8@8@1 and 3 8 3 = 8 @ 1,so that baryons which are bound states of 3 quarks belong to the singlet representation, which is totally antisymmetric as required by the Pauli principle and mesons which are bound states of qQ belong to the singlet representation which is totally symmetric. This assignment takes into account the fact that all known hadrons are color singlets. Thus the color is hidden. This is the postulate of color confinement and explains the non-existence of free quarks. Evidence for color also comes from 7ro 27 decay. Since 7ro is bound state of qQ i.e. lro)= IuU - dd), one can imagine that the decay takes place as shown in Fig. 1. The matrix elements M for the ro - decay, without, and with color [where we have to sum over the 3 colors for the quarks in the above diagrams] are
&
213
---f
Color, Gauge Principle and Quantum Chromodynamics
214
respectively proportional to
In fact the above quark triangle diagrams predict (7.la) where
without color ' with color
(7.lb)
and f n is the pion decay constant and is determined from the decay p+ + u, [see Chapter 111; its value is 132 MeV. Hence the decay rate is given by 7r+ -+
5,
With S, = this gives r(7r0-+ 27) = 7.58 eV in very good agreement with the experimental value rezp = 7.7460.50. Without color r t h will be a factor of 9 less in complete disagreement with the experimental value. Another evidence for color comes from measuring the ratio of e-e+ annihilation processes
R=
a(e-e+ + hadrons) a(e-e+ -+ p - p + )
-4
(7.3)
in the large center of mass energy fi = limit, where p l and p2 are the momenta of e- and e'+ respectively. To the
Evidence for Color
215
-c 1
-- e
+ -----
no
- 31 e
+
ki-
Y W
kz
Figure 1 Triangle diagrams for
T O ---t
27 through its constituents.
Color, Gauge Principle and Quantum Chromodynamics
216
lowest order in electromagnetic interaction, Eq. (A.77b) gives in the asymptotic region ( s >> rn;, m;)
Now for the inclusive process e-e+ -+ hadrons, we expect this to take place via e-e+ -+ qq and quarks (antiquarks) fragment into hadrons [see Fig. 21, so that (e-e+
---t
hadrons) = x a ( e - e '
---f
qij)
q
where the analogue of Eq. (4) gives in the asymptotic region [s >> m:,
41
o(e-e+
-+
47f 2 1 44) = --a[3eq]-, 3 S
(7.5)
where e, (in units of e) are the electric charges of the quarks which enter the photon-qq vertex [see Fig. 21 and the factor 3 arises because we have to sum over 3 colors for each quark flavor q. This gives in the asymptotic region
R =3 x e i . Q
For example, above the bottom quark threshold (see Chapter 8) i.e. for fi in the range 2mb < fi << r n [so ~ that weak interaction effects can be neglected],
6
> which is confirmed by experimental measurement of R above 2mb [see Fig. 31. Actually nature has also assigned a more fundamental role to color charges. We know that electromagnetic force is a gauge force; here we postulate that strong force is also a gauge force. In order to discuss the gauge force, we first state the gauge principle.
Evidence for Color
217
Y m
Figure 2
One photon exchange diagram for hadron production in
e-e+ annihilation.
5
4
R 3
oCRYSTALBAU
OJME
OMARKJ
+LENA *MAC
VMD-1
ITOPAZ
*PLUTO
WENUS
RTASSO
2
Em (Oev)
Figure 3
Compilation of R-values from different e-e+ experiments
Color, Gauge Principle and Quantum Chromodynamics
218
7.2
Gauge Principle
Suppose a physical system described by a wave function s ( x ) , x ( t ,r) has the property that under a phase transformation Q ( x ) + Q'(x) = eZ"*s(x)
f
(7.7)
(with A constant), the wave equation satisfied by @ or the corresponding Lagrangian is invariant. Now if we demand that it remains invariant when A is a function of space-time, then we shall show that it is necessary to introduce a vector boson which is coupled to a vector current with universal coupling e. We call such a phase transformation, local gauge transformation and the vector boson associated with it is a mediator of force whose strength is determined by the charge e. This is best illustrated by considering a non-relativistic particle of charge e and mass m described by a complex wave function e(x).Consider a space-time dependent phase transformation given in Eq. (71, with A as a function of x and e the electric charge. For this case the physical law is given by the Schrodinger equation 2 as v s=i-. 2m at
1 --
This is not invariant under the local gauge transformation (7). In order to restore gauge invariance, it is necessary to postulate a vector field A, = (4,A) and make the substitutions
V
a at
-+ +
V-ieA
a
.
+2e4
or
Equation (8) now becomes
1
-_ 2m_ (V - ieA)2@= i
(g
+zed)
Q.
(7.10)
219
Gauge Principle
This equation is invariant under the transformation (7), provided that A and 4 simultaneously undergo the transformations:
A
A+VA
or
A,
--t
A,
- aPA.
(7.11)
Ap 5 (4, -A) are the electromagnetic potentials. From the present point of view, the necessity for the existence of the electromagnetic potential A,(z) is a consequence of assuming invariance under the local gauge transformation. The electromagnetic fields E and B are related to the vector potential A, as follows:
B
=
VXA.
(7.12)
They are clearly invariant under the gauge transformations (1l), The Lagrangian density which gives Eq. (10) is given by
L = --1
2m
-e(p4 - j
+
A)
1 + -(E2 2 - B2),
(7.13)
where p = Q**,
1 e j = -(Q*VQ- (VQ*)Q)- -A@*Q. 22m 2m
(7.14a)
L is clearly invariant under the gauge transformations (7) and (11). p and j satisfy the equation of continuity
-aP +V.j=O.
at
(7.14b)
220
Color, Gauge Principle and Quantum Chromodynamics
This implies that the charge (7.14~) is conserved. Note also that the last term in Eq. (13) can be written ” , F,, = a,A, - &A, in manifestly covariant form - ~ F P V F ~where is the electromagnetic field tensor. The term -~F,,vF~U is the Lagrangian density for pure electromagnetic field. 7.2.1 Aharanov and Bohm elcperirnent We now discuss the question of testing the applicability of the gauge principle in electromagnetism. Taking the vector potential A to be independent of time and putting V = eg5> we try solution of Eq. (10) in the following form Q(r,t ) = Qo(r,t)eiTr
(7.15a)
lr
(7.15b)
where y(r) = e
A(r’) dt’.
Here Q can be regarded as a wave function of a particle that goes from one place to another along a certain route where a field A is present while Qo is the wave function for the same particle along the same route but with A = 0. It is easy to see that
Thus (15) is a solution of Eq. (10) when A(r) # 0 if e0((r,t) satisfies 1 2 0 a@o -v Q +VQO=i-(7.16) 2m at . The solution (15) has some striking physical consequences as shown in the two slit electron interferometer experiment proposed by Aharanov and Bohm [Fig. 41.
Gauge Principle
Figure 4 effect.
221
Double slit electron interferometer to test Aharanov-Bohm
In this experiment the magnetic field B (pointing in a horizontal direction out of the paper) is produced by a long solenoid of small cross-section and is confined to the interior of the solenoid so that the two electron beams (1) and (2) can go above and below the B # 0 region but stay within the B = 0 region and finally meet in the interference region P’. In the interference region, the wave function for the electron is
so that
where
+ e ilIp A(r’) dt’. P’
y1
= 7:
72 = 7:
+e
i2)p
(7.17b)
P’
A(r’) dt’.
(7.17~)
Color, Gauge Principle and Quantum Chromodynamics
222
Here 7: and 7: are the phases of the wave functions @! and !Pi in the absence of A. The interference pattern is determined by the phase difference
& A(r’) .
= $-$+e
=
dl .
S(B=O)+A,
(7.185~)
where C is the closed path PP’P and
A
=e
A(d) dl’ = e
s,
B d u = eiP.
(7.18b)
In Eq. (18) we have used Stokes theorem and put B = V x A and iP is the magnetic flux through the surface S bounded by the closed path C. Note the important fact that the phase difference A is gauge invariant while the individual phases y1 and 7 2 are not. Note also the remarkable fact that the amount of interference can be controlled by varying magnetic flux even though in the idealized experimental arrangement, electrons never enter the region B # 0. Now referring to Fig. 4 Phase difference - Path difference 21T x
where L is the distance of the screen from the slits. Thus from Eq. (18) we see that the diffraction maximum of the interference pattern for B # 0 is shifted from that for B = 0 by the amount Ay given by AY = eiP
(41) d 27r
(7.19)
This shift in the diffraction maximum, being gauge invariant, should be measurable. In fact the existence and magnitude of AharanovBohm effect has been confirmed to within 5% of the theoretical
223
Gauge Principle
prediction ( 19) by two qualitatively different experimental arrangements - one involving an electron biprism interferometer while the second used a Josephson-junction interferometer. The following comments are in order. (i) Measurement of Aharanov-Bohm effect not only varifies the gauge principle in electromagnetism but also quantum mechanics itself since classically the dynamical behavior of electrons is controlled by Lorentz force which is zero when the electrons go through magnetic field free region; yet in quantum mechanics observable effects are seen and depend on the magnetic field in a region inaccessible to the electrons. (ii) The vector potential A rather than the fields plays a crucial role as the basic dynamical variable in quantum mechanics. (iii) When A = 2n7r or
gauss
7.2.2 Gauge principle f o r relativistic quantum mechanics We now discuss the gauge principle for relativistic quantum mechanics. The spin 1 / 2 particle is described by Dirac equation with the Lagrangian density:
L = i%(z)2--yW,!P(z)- mi%(z)*(z).
(7.20)
In order that the Lagrangian density L be invariant under the gauge transformation (7), we must introduce a vector field A p ( z )satisfying Eq. (11) and replace in Eq. (20) 8,s by aPQ(z)--f
(a, +id,)@
= ope.
(7.21)
D, is called the co-variant derivative. The gauge invariant Lagrangian density is given by
1
L = ~ ( z ) i y p ( 8+, ieA,)Q - m@(z)!P(z) - -FpVFCu 4
Fpv = 8 , A U - &A,.
(7.22) (7.23)
224
Color, Gauge Principle and Quantum Chromodynamics
It is easy to see that under the transformation (11),FPvis invariant. Under the transformations (7) and (11), ~ , \ k + eieA(z)D P \k ,
(7.24)
so that GD,\k is gauge invariant, and so is m@\k.From Eq. (22), we see that the interaction of matter field Q with the electromagnetic field A, is given by
Lint = - e $ + P A ,
= -JfmA,,
(7.25a)
where
Jfm= e%'yP9,a,Jfm = 0
(7.2513)
is the electromagnetic current. We conclude that the gauge principle viz. the invariance of fundamental physical law under the gauge transformation gives correctly the form of interaction of a charged particle with electromagnetic field. To sum up the consequences of the electromagnetic force as a gauge force are as follows:
(i) It is universal viz. any charged particle is coupled with the electromagnetic field A with a universal coupling strength given by e, the electric charge of the particle. (ii) JZm is conserved. (iii) The electromagnetic field is a vector and hence the associated quantum, the photon, has spin 1. (iv) The photon must be massless, since the mass term p2APAPis not invariant under the gauge transformation. Thus unbroken gauge symmetry gives rise to long range force mediated by a massless gauge boson i.e. photon. (v) The covariant derivative 0, is an operator whose commutator is
Quantum Chromodynamics (QCD)
225
7.3 Quantum Chromodynamics (QCD) We now generalize the ideas of Sec. 2 to the case where there is more than one type of states, e.g. qa ( a = 1, 2, 3) and where there exist transformations [SUc(3)] between the different states
with (7.2713)
UU+
det U = 1
= 1,
and repeated indices imply summation. Here q a ( a = 1, 2, 3) for a particular quark flavor q form the fundamental representation of the color SU(3) group and XA, A = 1 . . . 8 , are the eight matrix generators of the group SUc(3) [see Chap. 5 for the form of these matrices. Although in Chap. 5 we discussed flavor SU(3) but the mathematics is the same]. Quarks are spin 1/2 particles. The Lagrangian density for free quarks (7.28a) L = FiY’apqa - qmqal where qa=
( i!)
andm=
(
mu
md
m s )
(7.28b)
is clearly invariant under the SU(3) transformation (27) with A constant. If we now require that the Lagrangian density (28) be invariant under the gauge transformation (27), with A(z) as function of spacetime, then as we have seen in Sec. 2, we must replace a, by its co-variant derivative which in the present case takes the form
D, =
i (a, - 2gsx G’>
=
(a,
JAG,+)
a -
(7.29)
Color, Gauge Principle and Quantum Chromodynamics
226
where gs is a scale parameter, the coupling constant and G Aare ~ vector gauge fields, their number being equal to the generators of SUc(3) group, namely 8. Then we note the important fact that the covariant derivatives satisfy the commutation relation
where in the matrix notation (7.31a) 1
1
-A*A(z) = -AAAAGA~ 2 2 1 G,, f s A . G,, = a,G, - 8,G, - ig, [G,, G,] = D,G, - DUG,
GA,,
=
(7.31b)
(7.31~)
GA, - &GA, 4- gsfABcGB,GcV (7.3Id)
1
z=
- G,,
2
*
G,,.
(7.31e)
Note the important fact that G,, in Eq. (30) provides the generalization of F,, [cf. Eq. (26) in Abelian case] for the present non-Abelian case. The two differ in the appearance of the last
Quantum Chromodynamics (QCD)
227
term in Eq. (31c) or (31d). This is because the gauge fields themselves carry color charges in contrast to photons which are electrically neutral in the electromagnetic case. Now if we replace the Lagrangian density (28a) by
or in the matrix notation by
(7.32b) then the Lagrangian density (32) is invariant under the infinitesimal gauge transformation [cf. Eq. (27b)l (7.33) provided that the vector fields GAPundergo the simultaneous transformation 1
G,
+
G,
+ i [A, G,] + LOPA
(7.34a)
ss
or GAP
GAp
- fABCABGCp
1
+ -gsa p A A *
(7.3413)
To see this, we note that under these transformations D,q
+
(1
+ ;A
*
A(+)] D,q
(7.35a)
It is then trivial to show that the Lagrangian (32) is gauge invari-
ant.
228
Color, Gauge Principle and Quantum Chromodynamics
For the finite gauge transformation (27), we have the gauge invariance provided that the gauge fields G, simultaneously undergo the transformation (7.36) Under these transformations:
and hence the Lagrangian (32) is gauge invariant. ~ called gluons. They The eight gauge vector bosons G A are are mediators of strong interaction between quarks just as photons are mediators of electromagnetic force between electrically charged particles. The gauge transformation given in Eq. (27) is called the non-Abelian gauge transformation, whereas the gauge transformation (7) is called the Abelian gauge transformation. The non-Abelian gauge transformation was first considered by Yang and Mills and gauge bosons are sometimes called Yang-Mills fields. 7.3.1 Conserved c u r r e n t In order to discuss the conserved current associated with gauge fields, we discuss a general method. Suppose we have a set of fields which we denote by The Lagrangian is a function of these fields $a and L = L($a,8,$a). (7.38) Consider an infinitesimal gauge transformation
4a(x)
--+
@a(x)+ ~AA(~)(TA):$~. (7.39)
TAare matrices corresponding to the non-Abelian gauge group and the representation to which the fields $ a ( z )belong. From Eq. (38),
Quantum Chromodynamics (QCD)
229
Using the Euler-Lagrange equations
(7.41) and the fact that
S(apda)= apS(q5,),we have
(7.42) On using Eg. (39) so that
= iAA(TA):$b,
we get
(7.43)
If we take AA as constant i.e. independent of 2,then we can rewrite Eq. (43)as
Hence we have the Noether's theorem. If the Lagrangian is invariant under the gauge transformation (39) with constant A A , i.e. SL = 0, then the current given in Eq. (45) is conserved. Let us apply this to the QCD Lagrangian (32). Here $a correspond to G B pand qa. Now, for the gauge vector bosons which belong to the adjoint representation of SUC(3), we have i(T~)g = - ~ B A Cand for the quarks which belong to the triplet
230
Color, Gauge Principle and Quantum Chromodynamics
representation of SUc(3), TA = ~ A A .Then using the expression (31d) for GAP”in the Lagrangian (32), Eq. (45)gives
(7.46)
The current F i is universally coupled to the gauge fields G A with ~ universal coupling g s . Now the interaction part of the Lagrangian (32) is given by
(7.47)
The last term of Eq. (47) represents the self interaction of gauge bosons among themselves as they carry the color charges. This term is very important in QCD and is responsible for the asymptotic freedom of QCD. From Eq. (47), the q q G, G G G and G G G G vertices in the momentum space can be represented graphically as shown
231
Quantum Chromodynamics (QCD)
below
h
i98Y
(+)
The Feynman rules for the QCD Lagrangian are discussed in Appendix B. 7.3.2 Experimental determinations of a, (q2) and asymptotic freedom of QCD The important physical properties of QCD are (i) the gluons, being mediators of strong interaction between quarks, are vector particles and carry color; both of these properties are supported by hadron spectroscopy discussed in the next section, (ii) asymptotic freedom which implies that the effective coupling constant a, = g:/47r decreases logarithmically at short distances or high momentum transfers, a property which has a
232
Color, Gauge Principle and Quantum Chromodynamics
rigorous theoretical basis. This is the basis for perturbative QCD which is relevant for processes involving large momentum transfers, (iii) confinement which implies that potential energy between color charges increases linearly at large distances so that only color singlet states exist, a property not yet established but find support from lattice simulations and qualitative pictures (see next section) and from quarkonium spectroscopy to be discussed in Chap. 8. In this section, we discuss the present evidence for QCD being asymptotic free. First we note that due to quantum radiative corrections, a, evolves with the characteristic energy of the process in which it appears. Actually these corrections give
*I
,
(7.48a)
where X2 >> Q2 and must be introduced so that the integrals involved in these corrections are convergent. Here . ' . denotes higher order corrections and 0 is the momentum carried by a gluon at quark-quark-gluon vertex which defines g s ( Q 2 ) .It is convenient to rewrite Eq. (48a) as
(7.4813)
(7.48~)
We now eliminate the unobserved " bare" coupling constant a , and ~ the cut-off X2 by making a subtraction at Q2 = p2. Thus we obtain a,-1 ( Q 2 )- a ; ' ( p 2 ) = bln-Q 2 P2
(7.48d)
233
Quantum Chromodynamics (QCD)
with b = 87rbo. Or 1
as(Q2) =
+
a ~ ~ ( pb In~Q)2 / p 2
(7.48e)
The constant b is evaluated in Appendix B and is given by 1
b = - (11 47r
-
2 gnf)
(7.48f)
where nf is the number of effective quark flavors. Another way of writing Eq. (48e) is
ai1(Q2)= bln-
Q2
A&D
where
a,-1 ( p ) - blnp' = -blnA&D.
Thus finally we have (7.48h) and we see the running of a,(Q2)with Q2. A Q ~ Dis the QCD scale factor which effectively defines the energy scale at which the running coupling constant attains its maximum value. AQCD can be determined from experiment. For gnf < 11, it is clear from Eq. (48b) or (48h) that a,(Q2)decreases as Q2 increases and approaches zero as Q2 --t 00 or T- + 0. This is known as the asympotic freedom property of QCD.This is due to the factor 11 in Eq. (48f) or (48h) and arises due to the self-interaction of gluons (see Appendix B). We now discuss the experimental determination of the coupling constant a,(Q2) at various values of Q2 from different reactions, starting from the lowest value of 0.
Color, Gauge Principle and Quantum Chromodynamics
234
The rates of quarkonium decay, in particular the ratio of [see, in particular Eq. (8.45)] provides a deterrates r,p/l?ggg at rnJ/,,,and rnr for the charmonium and mination of a, bottomonium states and give
(m)
"3(mJ/$)= 0.216 k 0.024, a , ( m ~=) 0.178 f 0.005
(7.49a)
where we have used
R,
( '4 )
r =
(
T
-+
hadrons
The value of a, obtained from the scaling violations in deep inelastic lepton-nucleon scattering [see Chap. 141 gives a,
(@
= 2.6
GeV = 0.264
0.101.
(7.4913)
The order a, corrections to the total hadronic cross-section in e-e+ annihilation in the ratio (3) modify it from Eq. (6) to
By fitting the value of R at 4 = 34 GeV shown in Fig. 3 one obtains ~ ~ ( GeV) 3 4 = 0.142 f 0.03. (7.49c) Finally from the semi-leptonic branching ratio R, for the inclusive decay T -+ u, hadrons, one obtains
+
a,(?%)= 0.35 f 0.03.
235
Quantum Chromodynamics (QCD)
(in 0.04
0.1
I'
'
scheme, in GeV) 0.5
0.2
"
--c
Average e+e- rates "
e+e- event shapes
__o__
-
Fragmentation Z width
-0-
Small x structure functions ep event rhapes c
Deep Inelastic S c e g (DIS) PoluizedDIS t decays
=
QQ L a t t i c G rdpy 0.10
0.12
0.14
a,(Mz)
Figure 5 Summary of the values of a , ( m ~ )and A(5) from various processes. The values shown indicate the process and the measured value of as extrapolated upto p = mZ. The error shown is the total error including theoretical uncertainties.
Figure 5 shows the values of cr,(m,) deduced from the various experiments. Figure 6 clearly shows the experimental evidance for the running of ~ , ( p i.e. ) decrease of the coupling constant as increases as indicated by Eq. (48). An average of the p = values in Fig. 5 gives Qs(rnz)= 0.119 f 0.002
which corresponds to
(7.50) The LEP / SLC value for a,(rnz) is 0.124 f 0.004.
Color, Gauge Principle and Quantum Chromodynamics
236
0.4
0.3
am 0.2
0.1
0.0
1
2
6
10
20
50
100
200
1 (GeV)
Figure 6 Summary of the values of a,(p) a t the values of p where they are measured. The figure shows clearly the decrease in a s ( p ) with increasing p.
Hadron Spectroscopy
Figure 7 system.
7.4
237
Diagram generating one-gluon exchange potential for q?j
Hadron Spectroscopy
7.4.1 One gluon exchange potential All known hadrons are color singlets. Just as an exchange of photon gives force of repulsion between like charges and force of attraction between unlike charges, the exchange of gluon gives force of attraction between color singlet states. The exchange of gluons can provide binding between quarks in a hadron. For qij system (meson), the color electric potential due to one gluon exchange diagram [see Fig. 71 is given by:
The factors L6' and $6: in the initial and final states arise due to normalize color singlet totally symmetric wave function for the qq system. The minus sign arises due to the coupling of a vector particle to the antiquark. Here i ,j are flavor indices and a, b, c, d are color indices. Since T T ( X A X , ) = T r ( X A X A ) = 16, we get
ea
(7.52)
Color, Gauge Principle and Quantum Chromodynamics
238
Figure 8 Diagram generating one-gluon exchange two-body potential for three quarks (baryon) system.
For three quarks system (baryon), one gluon exchange diagram (Fig. 8) gives the following two-body potential
3
The factors and arise due to the fact that three-quark color wave function is totally antisymmetric in color indices. Using Eebd - StS,d - S,dS,b, and T ~ X= A0, we get
(7.5313) Note the important fact that in both cases, we get an attractive potential. We also note that V,? = 2x39 for color singlet states. Thus we can write the two-body one-gluon exchange potential as
(7.54) Since the running coupling constant as becomes smaller as we decrease the distance, the effective potential Kj approaches the lowest order one-gluon exchange potential given in Eq. (53) as T + 0.
Hadron Spectroscopy
239
Now in momentum space, we can write the potential in QCD perturbation theory for small distances (. < 0.1 fm) as
(7.55) where V(T) is the Fourier transform of V(q2) and q2 is the momentum conjugate to T-. The running coupling cys(q2)in QCD is given by Eq. (48h). We conclude that for short distances, one can use the one gluon exchange potential, taking into account the running coupling constant as(q2). 7.4.2 Long range QCD motivated potential The second regime, i.e. for large T-, QCD perturbation theory breaks down and we have the confinement of the quarks. Thus unlike the short range part of the potential, the long range part cannot be calculated on perturbative QCD as the QCD constants become large in this region. Perturbative QCD gives no hint of intrinsically nonperturbative phenomena such as color confinement. One may look for the origin of this yet unsatisfactorily explained phenomena. There are many pictures which support the existence of a linear confining term. One of these is discussed below:
The string picture of hadrons: This picture is depicted in Figs. 9 and 10. A string carries color indices at its ends. Gauge invariance implies that each site must be a color-singlet. Thus, an allowed configuration of a quark and an antiquark on adjacent sites is the one in which the quark and antiquark are linked by a string so that the color index of quark (antiquark) and the color index of the string at that end are contracted to form a color singlet. When a quark and an antiquark are far apart, many strings have to be excited to connect the two sites [see Fig, lo]. When there is enough energy available to create a new qq pair, the system breaks up permitting the formation of two color singlets. Calculation based on this theory shows that the energy stored in this configuration is:
240
Color, Gauge Principle and Quantum Chromodynamics
Figure 9
String picture of qq.
Figure 10 String separation of a quark-antiquark pair.
L E = Toa
for L
>> a,
where L is the quark-antiquark separation and TOis the string tension. To isolate a quark for example, the antiquark in the above illustration has to be removed to infinity; it clearly takes an infinite amount of energy to do this. This is the basis of color confinement. The confining potential is of the form:
V(T) N constant x
T,
for T > 1/M, where M is a typical hadronic mass scale. Thus $ is of order of the hadron size of 1 fm = 5 GeV-' so that M M 200 MeV. The confining potential is spin and flavor independent. This picture is supported by the observation that hadrons of a given internal symmetry quantum number but different spins obey a simple spin ( J ) - mass ( M ) straight line relation i.e. we say that they lie on linear Regge trajectories, an example of which is displayed in Fig. 11.
241
Hadron Spectroscopy
spin
h 1 2 Figure 11 Regge trajectories for non-strange (I = 1) and strange ( I = 1/2) bosons.
For the families of hadrons composed entirely of light quarks, the above mentioned relation between J and M 2 for Regge trajectories is given by: (7.56a) J (M 2 ) = QO Q’M’,
+
with
a’
“N
0.8 - 0.9(GeV/c2)-’.
(7.56b)
The connection between linear energy density and the linear Regge trajectory is provided by the string model formulated by Nambu. We consider a massless (and for simplicity spinless) quark and antiquark connected by a string of length T O , which is characterized by an energy per unit length 0. The situation is sketched
242
Color, Gauge Principle and Quantum Chromodynamics
below:
For a given value of length ro,the largest achievable angular momentum J occurs when the ends of the string move with the velocity of light. In these circumstances, the speed at any point along the string at a distance r from the center will be: ( p = V/C)
The total mass of the system is then:
M=21
dra
7r
(7.57a)
JW- 2
= arg-,
while the orbital angular momentum of the string is: TOPdr a rP(r)
J=2L
27r
JW= aro-8 .
(7.57b)
Using the relation (57a), one finds that: J=-
M2
(7.58a)
27ra ’
which corresponds to a linear Regge trajectory with (7.58b) This connection yields: 0.18 GeV2 0.20 GeV2
ora =
0.9 GeVP2 0.8 G e V 2
(7.59) ’
Hadron Spectroscopy
243
This heuristic estimate of the energy density suggests that at a separation of the order of 1 fm, we may characterize the interquark interaction by the linear potential
V ( r )= ur.
(7.60)
The lattice gauge theory calculations also support the linear form for the long range part of the QCD potential, Thus phenomenological potential of the form (7.61) can be used for heavy quarks. The Cornell potential
K V ( r )= --r
r ++ c, a2
(7.62a)
where
K
= 0.48, a = 2.34(GeV)-'
and C = -0.25
(7.62b)
has been used successfully to describe mass spectrum of charmonium and bottomonium systems [see Chap. 81. Note that value of a (s-)-) in Eq. (62b) is consistent with the value of u stated above [cf. Eq. (59)]. The purely phenomenological potentials of the form and ~ ( r=)a + bro.l (7.63a) and
V ( r )= C l n r
(7.63b)
have also been used successfully for cF and bb systems. 7.4.3 Spin-spin interaction Finally, we note that a spin 1/2 charged particle of charge eQi has a magnetic momentum e,ui = $oi. In quantum mechanics, the energy splitting between S-states (zero orbital angular momentum) is given by two-particle operator (Fermi contact term) (7.64)
244
Color, Gauge Principle and Quantum Chromodynamics
Similarly in QCD, we have eight color-magnetic moments
The analogous two-particle interaction for QCD is then given by
(7.66) Again for a color singlet system
(7.67) Eq. (67) would immediately give m(3S1) > m('So) [for example mp > m,] in agreement with the experimental result. This supports the fact that gluons are spin 1 particles.
7.5
The Mass Spectrum
The one gluon exchange potential is obtained by summing over all possible quark indices in Ky in a multiquark system like qij and qqq. Thus
=
CyF.
(7.68)
i>j
The potential Vc for S-states is found to be [in non-relativistic limit keeping terms up to (p2/m2)]
The Mass Spectrum
245
(7.69) The first term on the ri ht hand side is the potential in the extreme non relativistic limit = 0); spin dependent term is due to the color magnetic moments interaction as mentioned previously. For S-states,
f
Now our Hamiltonian, including the rest masses of the quarks can be written as
where
p- 2. = - t i 2 0 2
(7.72)
Here Vc(r)is the confining potential, VG(T) is the one gluon exchange potential given in Eq. (69), i is the quark flavor index, i.e. i = u , d , s for ordinary hadrons. We will take mu = m d . In order to discuss the mass spectrum of hadrons, we have to take the expectation value of the Hamiltonian H(r) with respect to the relevant wave functions of the hadrons. The wave function is the product of three parts viz. unitary spin, spin and space parts. For s-wave, we write the space function as Qs(r). Let us first take the expectation value of H(r) with respect to Qs(r),we have
(7.73)
Color, Gauge Principle and Quantum Chromodynamics
246
where (7.74a) (7.7413) (7.74c) (7.74d)
Note that the mass operator M is still an operator in unitary spin and spin space. The parameters a , Ao, d , b and c may be different for L = O(qq) meson and L = O(qqq) baryon systems. We first apply the mass formula (73) to pseudoscalar meson system.
7.5.1 Meson mass spectrum From Eq. (73), the mass operator for S-wave mesons can be written as
(7.75) where
(7.76)
Indices 1 and 2 refer to the constituent antiquark and quark respectively. For vector gluon k , = Now
-4.
s1 * s 2 =
{ 's
spin triplet state S = 1: vector meson spin singlet state S = 0: pseudoscalar meson.
The Mass Spectrum
247
-$,
as for vector gluons, it is clear from Eqs. (75)
Thus if k, = and (76) that
m(3S1)> m(lS0)
(7.77)
i.e. vector meson mass is greater than the corresponding pseudoscalar mass in agreement with experimental observation. If gluons were scalar particles, then s1 s2 term would be absent so that m(3S1) = rn('S0) in disagreement with the experimental observation. For pseudoscalar gluons, lc, = since pseudoscalar coupling is the same for antiquarks. In this case we would have m(3S1)< m(lSo),again in disagreement with the experimental result. We conclude that the experimental results about meson spectrum support the fact that gluons are vector particles and are thus quanta of QCD. From Eq. (75), we can write down the masses of vector and pseudoscalar mesons. For example (with mu = md): e
i,
For K' and K , replace 2mu by m,
+ m,, 2/mu by (k+ k),
(A+ $)
and 2/m: by in rnp and m, respectively. mp = m, and for rn4 replace mu by m, in the expression for mp. From Eq. (78), we have the following results 1 b Y'
mumS
m, mp-m, mk-mK m&- mK mp - m7r
=
mP
(7.79)
64 1 -asdT 3mE 9 mu 1 16 - 64 = d = -a,d3mum, 9 mums mu - M 0.66 (Expt 0.64), m8 16
= -d
-
(7.80) (7.81) (7.82)
248
Color, Gauge Principle and Quantum Chromodynamics
where we have used for mu and m,, the values of constituent quark masses [mu= 336 MeV, m, = 510 MeV] obtained for the magnetic moments of baryons (see Chap. 6). We also obtain
(7.83)
z:;:;
where X = is the SU(3) symmetry breaking parameter. Hence to order A, we recover the Gell-Mann-Okiibo mass formula with ideal mixing angle between w8 and w1. For pseudoscalar mesons qns and q,, we get mqns - mlr mvs = (2mK
- m,)
+ 0(X2).
(7.84a) (7.84b)
These formulae are badly broken. Thus the above analysis breaks down for J = 0 mesons, q and 7'. The reason for this is that our Hamiltonian does not take into account quark-antiquark annihilation into gluons. The lowest order annihilation diagram is shown in Fig. 12. This diagram contributes only to 'So state, because of charge conjugation conservation. Since gluons do not carry any flavor, therefore it contributes to I = Y = 0, 'SOstates only. This diagram is relevant only for 9 and q' mesons, and is of order O(a:). For I = Y = 0 vector bosons, the diagram with threegluon exchange contributes, which is of order O ( a i )and hence can be neglected. We now take into account the diagram of Fig. 12 for pseudoscalar mesons. If ua, d d and SS can annihilate with an amplitude A, which we assume to be SU(3) invariant, then there will be an additional contribution to the mass matrix, which in the u U , dd
The Mass Spectrum
249
Figure 12 The qij annihilation diagram for 'So state through two gluons.
and
SS
basis is given by
A A A (7.85)
A A A Taking into account Eq. (84) and the fact that lqs) = IsS), Iqns) = I(u0 + d z ) ) , IT' ) = I(ua - dd)), we get in no, qns and qs basis, the mass matrix
&
&
0 0
0
0
m,+2A fiA
JZA 2m~-m,+A
).
(7.86)
From Eq. (86),we note that we have to diagonalize the mass matrix
M
+M
+
Man, =
). ( m$jA 2mK .\/ZA m, + A -
(7.87)
For this purpose, we define the physical states as (see Problem 5.15) 17) = cos 4 Iqns) - sin $1175) 177') = sin 4 Iqns) C O 4 ~ 17s)
+
*
(7.88)
250
Color, Gauge Principle and Quantum Chromodynamics
Then the mass eigenvalues are given by mqmqt = m,(2mK
+
- m,)
+ A(4mK - m,)
+
mq+mm,/= mqns m,, = 2 m ~ 3A.
(7.89)
Using the experimental values for q and q’, we can determine A. The mass scale A comes out to be x 172 MeV, a rather low value compared to mq and mqt which is both interesting and reasonable. To conclude, we have shown that mass spectrum of vector mesons can be explained successfully. With the addition of annihilation diagram, the pseudoscalar meson mass spectrum can also be understood. 7.5.2 Baryon mass spectrum In order to discuss the mass spectrum of the baryons, it is convenient to first calculate the matrix elements of the spin operator 1
= zmimjsz * sj
(7.90)
between spin states. The eigenvalues of si . s j are 1/4 and -3/4 for spin triplet, and singlet states respectively. Therefore, 1
si * sj
ITT)
=
4 ITT)
si *
ILL)
=
4 ILL)
sj
1
(7.91a)
(7.91b)
From Eq. (91), we get si sj I z T j L )
=
sz ’ sj I z L j T )
=
*
-41 lzy)+ 1 I z y ) 1 1 -- l z y ) + p j l ) . 4
(7.92)
The Mass Spectrum
25 1
The spin wave functions for baryons are given in Table 6.3 and Eq. (6.8). Using these wave functions, we get with the help of Eqs. (91) and (92) for baryons with s, =
;+
=
c
-sz
1 *
sj
i> j mimj
Juud)
4:
(--A) I(tI + IT) T
-2lTt.l.)
1
3 4m2
= --+p).
(7.93a)
Similarly we get (7.9313)
4 4
(7.93c) (7.93d)
where we have used
(7.94)
Color, Gauge Principle and Quantum Chromodynamics
252
%’
For baryons, we take s, of O,,. Now
= 3/2
and calculate the matrix elements
(7.95a) Similarly we get
R,,
p*+>
= 1
as,lp*o)
=
n,,(n-)
=
(-
2 4 mums
+
5)[c*+)
(7.9513) (7.95c)
+r), 4 m:
(7.95d)
where we have used (7.96a) (7.96b) (7.96~) Since the spin-spin interaction term from Eqs. (73) and (90) is, 3
(7.97)
we have from Eqs. (93) and (95):
in agreement with experimental observations. For gluons with color, k, = -2/3; if gluons do not carry color, then k, = 1 instead of -2/3 and we would get results in contradiction with experimental values. This supports that the vector gluons carry color.
The Mass Spectrum
253
The spin dependent term R,, splits the masses of baryons with the same quark content, but with different spin. Thus, we get from Eqs. (93), (95) and (97):
rna-m, mc-mA
d 8m2 16 d = -- (1 3 me =
2) (7.98)
where d =
y. From Eqs. (98), we get
mz* - mz me8 - me 2mc. mc - 3 m 2 (mA- mp) mc-mA 2 = - (1 mA-mp 3
+
1
(expt 1.12) (7.99a)
=
1
(expt 1.04) (7.9913)
=
0.23
(expt 0.26). (7.99~)
=
~
2)
In the above derivation, the effects of wave function distortion due to symmetry breaking by quark effective masses have been neglected. These effects will give slight deviations from unity in the relations (99a, b). We now discuss the baryon masses of same spin, using Eqs. (73), (93) and (98). We can write the baryon mass formula:
m = (ml+rn2+ms)+a
+36
1 1 + c (-m1m2 +m2m3
1
m3m1 (7.100)
254
Color, Gauge Principle and Quantum Chromodynamics
where C = ia,c, 6 = - ~ a , b . Introducing the SU(3) breaking parameter X = (m, - m,)/(m, mu)and writing mo = f(m, mu) and retaining only the first order term in A, we get
+
+
mp = A + X mA = A + X mL = A + X V I , ~=
A+X
(7.101)
where
mo+b+-+$-+-a 'd)+A0 mo mo 3 m i
(7.102)
From Eq. (101), we get the Gell-Mann-Okubo mass formula rn,+mz - mc+3m~ 2 2
(7.103)
We conclude that both the meson and baryon mass spectra can be explained quite well in QCD. In this simple picture, we have used non-relativistic quantum mechanics for u , d and s quarks. Although this approximation is not so good for these quarks (a their masses are less than 1/2 GeV) and at this energy scale QCD perturbation theory may not be a good approximation, even then the results are good.
Bibliography
7.0
255
Bibliography
A. General 1. E. Abers and B. W. Lee, Gauge Theories, Phys. Rep. 9C, 1 (1973). 2. B. W. Lee, ”Particle Physics”, in Physics and Contemporary Needs, Vol. 1 (Ed. Riazuddin), 321, Plenum Press, New York (1977). 3. K. Huang, Quarks, Leptons and Gauge Fields, World Scientific, Singapore (1982). 4. K. Moriyasu, An Elementary Primer for Gauge Theory, World Scientific, Singapore (1983). 5. C. Quigg, Gauge Theories of the Strong, Weak and Electromagnetic Interactions, Benjamin/Cummings, Reading Massachusetts, (1983). 6. T&Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics, Clarendon Press, Oxford (1984). 7. M. Chaichian and N. F. Nelipa, Introduction to Gauge Field Theories, Springer-Verlag (1984). 8. T. D. Lee, Particle Physics and Introduction to Field Theory (revised edition), Harwood Academic, New York (1988). 9. J. J. R. Aichison and A. J. G. Hey, Gauge Theories in Particle Physics (2nd edition), Adam Hilger, Bristol, England (1988). 10. C. H. Llewellyn Smith, Particle Phenomenology: The Standard Model, OUTP-90-16P, The Proceedings of the 1989 Scottish Universities Summer School: Physics of the Early Universe. 11. R. E. Marshak, Conceptual Foundations of Modern Particle Physics, World Scientific (1992). 12. M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, Mass (1995).
B. For Sec. 7.2.1 1. D. Bohm and H. J. Hiley, I1 Nuovo Cimento 52A, 295 (1979).
Color, Gauge Principle and Quantum Chromodynamics
256
C . QCD 1. E. Reya, Perturbative Quantum Chromodynamics Phys. Rep. 69 C, 195 (1981). 2. A. H. Mueller, Perturbative QCD at High Energies, Phys. Rep. 73 C, 237 (1981). 3. G. Altarelli, Partons in Quantum Chromodynamics, Phys. Rep. 81 C, 1 (1982) 4. F. Wilczek, Quantum Chromodynamics: the Modern Theory of the Strong Interaction, Ann. Rev. Nucl. and Part. Sci. 32, 177 (1982). 5. D. W. Duke and R. G. Roberts, Phys. Rep. 120, 275 (1985). 6. T. Muta, Foundation of Quantum Chromodynamics, World Scientific, Singapore (1987). 7. R. D. Field, Applications of Perturbative QCD, Addison-Wesley (1989). 8. Perturbative Quantum Chromodynamics, Editor: A. H. Mueller, World Scientific, Singapore (1989). 9. M. Creutz, Quarks, Gluons and Lattices, Cambridge University Press (1983). 10. Ref. 12 in A above
D. For Sec. 7.3.2 1. G. Altarelli, Experimental Tests of Perturbative QCD, Ann. Rev. Nucl. and Part. Sci, 39, 357 (1989). 2. Ref. 1 2 in A above
E. For Figs. 3 and 5 and 6 1. Particle Data Group, The European Phys. J. C3, 1-4 (1998).
F. Hadron spectroscopy 1. A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev. D12, 147 (1976). 2. B. W. Lee, Ref. 2 in A above. 3. 0. W. Greenberg,
Bibliography
257
Ann. Rev. Nucl. Part. Sci. 28, 327 (1978). 4. F. E. Close, An Introduction to Quarks and Partons, Academic Press, New York, 1979. 5. C. Quigg, "Models for Hadrons", in Gauge Theories in High Energy Physics, edited by M. K. Gaillard and R. Stora (Les Houches, 1981), North-Holland, Amsterdam, 1983, p. 645. 6. J. L. Rosner, "Quark Models", in Techniques and Concepts of High Energy Physics (St. Croix, 1980), edited by T. Ferbel, Plenum, New York, 1981.
G . For Sec. 7.4.2 1. B.W. Lee, Ref. 2 in A above. 2. C. Quigg, Quantum Chromodynamics near the Confinement Limit, FERMILAB-Conf-85/126-T (1985). 3. Y. Nambu, Phys. Rev. D10, 4262 (1974). 4. S. Mandelstam, In Proc. 1979 Int. Sym. on Lepton and Photon Interaction at High Energies (ed. T. B. W. Kirk and H. D. I. Abarbanel). Fermi Lab., Batavia, Illinois (1979). 5. S. Gasiorowicz and J. L. Rosner, Am. J. Phys. 49, 954 (1981).
Chapter 8 HEAVY FLAVORS 8.1
Discovery of Charm
The J/\k was discovered in 1974 in the reaction
p+Be-e'e-+X at fi = 7.6 GeV. A narrow peak at rn(e+e-) = 3.1 GeV was found. It was also seen in e+e- collision at fi = 3.105 GeV in the following reactions e-e+ e-e+ e-e+
---+
e-e+ p-p+
hadrons.
The width of the resonance was very narrow. It was less than the energy spread of the beam, r 5 3 MeV. For this reason, the width cannot be read off directly from resonance curve. The resonant cross section for any final state f :
e-e'
--+
J/Q
-+
f
is given by the Breit-Wigner formula [cf. Eq.(4.52)]:
259
Heavy Flavors
260
where J is the spin of the resonance, m is its mass, is the spin of electron or positron and
s1
= sg = 1/2
Here k = IkJis the center of mass momentum. r is the total width, I?, and rf are the partial widths into e-e+ and f respectively. We can write Eq.(l) as
Since the resonance is very narrow, r is very small and it is a good approximation to replace the denominator in Eq.(3) by the 6-function $V(& - m ) and then integration of Eq. (3) gives
Now Cf oef = otot,CfI'p = the process
and assuming I?, = F p , we have for
We also have for the total cross section
Assuming the spin of the resonance J / @ , J = 1, we determine the widths re = rp and the total decay with r. Since r = re rlL r h , we can also determine the hadronic decay with I'h. The experimental values for these decay widths are given below:
+
+
m(J/\k) = 3096.8850.04 MeV, re = rp= 5.26 f 0.37 keV, r = 8 7 & 5 keV.
Discovery of Charm
261
The J / 9 spin-parity can be determined from a study of the interference between e-e+ -+ y + p-p+ and e-e+ -+ 9 -+ p-p+. The cross section for the QED process e-e+ y --t p-p+ is well known [Eq. (78) of Appendix A] and is given by
-
(s
>> m:,
m;)
da Q2 _ -- -(I
dR
4s
+ C O S ~e)
If the spin-parity of J / @ is that of photon viz. I-, then the angular distribution would not change by the interference between QED amplitude and the resonant amplitude. In fact, experimentally, it was found to be (1 cos20 ) near the resonance, clearly establishing the spin-parity of J / 9 to be 1-.
+
8.1.1 Isospin
Experimentally the decay J / Q -+ p p occurs with a branching ratio Fpp/r = (0.214 f O.OlO)%, which is too large to be explained by the electromagnetic effects. Now p p can have only I = 0 or I = 1. Thus the isospin of J / Q is either 0 or 1. If J / Q has I = 1, then the decay J / @ + po7ro is forbidden while for I = 0 (see problem l),we have
to be compared with the experimental value of 0.494 f0.068. Thus the isospin of J / Q is 0. Now G-parity is given by G = (-l)'C, where C is the charge conjugation parity of J / Q . Since I = 0, therefore, G = C. The allowed decay J / 9 --t poxo fixes its Cparity to be C = (-l)(+l)= -1. Hence G = -1 for J / @ . 8.1.2 SU(3) classification Due to C-invariance, the VPP coupling is F-type (see Sec. 5.8.2) which is not possible if V is an SU(3) singlet. Thus an SU(3)
Heavy Flavors
262
singlet vector meson cannot decay into two pseudoscalar mesons belonging to the same SU(3) multiplet. In particular if J / Q is an SU(3) singlet, then J / 6 -+ K K is forbidden while J / Q --t K*K or J/\k 4K * K is allowed by C-invariance. Experimentally one finds
which shows that J / 6 is an SU(3) singlet. If J / Q is an SU(3) singlet, then its invariant coupling with PV is given by
\k,TT(PV)
=
Q[ pOnO
+ p - r + + p+n- + fT*-I(+ + K * + K (8.10)
(J'Q r(
Hence we have
--t
J/Q
+ KK*)
=
1 (phase space correction)
=
1.2
(8.11)
to be compared with the experimental value 1.3910.12. To summarize, the J / 6 resonance is an SU(3) singlet. with J p c = 1--, G = -1 and I = 0. 8.2
Charm
Although J / Q itself does not carry any new quantum number, its unusually narrow width in spite of large available phase space suggests that it is a bound state of cC , where c is a quark with a flavor which is outside the three flavors u, d and s of SU(3). This new flavor is called charm. The quark c is assigned a new quantum number C = 1 and C = 0 for u,d and s quarks. Thus to take this quantum number into account, the Gell-Mann-Nishijima relation would be modified to 1 (8.12) Q = 1 3 2(Y C ) .
+
+
263
Charm
Figure 1
allowed by 021
For the charmed quark c, C = 1, 13 = 0, Y = B = 1/3. Thus the charge of charmed quark is 2/3 and its mass rn, FZ ~ V L J ,=~ 1.55 GeV. (87 @ keV compared to 100 MeV The narrow width of ,I/ for p ) can be qualitatively understood by the 021 rule, just as the suppression of 4 --f 37r compared to 4 3 K K is explained (see Sec. 5.5.9) by this rule. Thus the decay depicted in Fig. 1 is allowed but that shown in Fig. 2 is suppressed by 021 rule. But the decay J / $ D D shown in Fig. 1 is not allowed energetically since mJpp < 2 m ~ . --f
8.2.1
Charmed mesons
The charmed quark c can form bound states with Q, where q = u, d, s. The low lying bound states such as cq have been found experimentally and are listed in Table 1. The C = 1 states ( D + , Do)D,’ form an SU(3) triplet (3); (D+, Do) form an isospin doublet. Similarly C = -1 states q C : (DO, W ) D ; form an SU(3) triplet (3). The states D*and 0; are unstable and decay strongly and
Heavy Flavors
264
C
.t-> C
Figure 2
J/$J suppressed by 021 rule. Table 8.1
Charm
265
Table 8.2
JP 0001112+ or 1+ 2+ or 1+
radiatively. For example
8.2.2 The fi'h quark flavor: Bottom mesons Fifth quark was discovered, when in 1977 the upsilon meson T(J p c = 1--) was found experimentally as a narrow resonance at Fermi Lab. with mass N 9.5 GeV. This was later confirmed in e+eexperiments at DESY and CESR which determined its mass to be 9460 f 10 MeV and also its width. The updated parameters of this resonance [from the Particle Data Group Tables] are mass 9460.37 f 0.21 MeV and width 52.5 f 1.8 keV. Again the narrow width in spite of large phase space available suggests the existence of a fifth quark flavor called beauty, with a new quantum number
Heavy Flavors
266
B = -1 for the bottom (b) quark. With this assignment the formula Q = I3 1/2(Y B C) would give the charge of b quark the value -1/3(13 = 0). The mass of b quark. is expected to be around 4.9 GeV as suggested by the Y mass which is regarded as a 3S1bound state of bb. Thus one would expect particles with B = *l, such as bq or qb. The lowest lying bound states bq and qb have been found experimentally, they are given in Table 2. The B = -1 states (Bo,B-)B: form an SU(3) triplet (3) and B = +1 states ( B + ,B0)B: form another triplet (3). 8.2.3 The sixth quark flavor: The top The top quark t with Q = 2/3 and new flavor T = 1 was expected on theoretical grounds. It was first found experimentally in 1996; its mass is mt = 175 f 6 GeV. Since (t, b) form a weak doublet, it decays weakly to W+ b, i.e.
+
+ +
+
t
3
w++ b.
The predicted decay (see Eq. (15.27)) rate is
where we have neglected the b quark mass compared to mw and mt. Taking m, = 175 GeV, m w = 80 GeV, GF = 1.166 x GeVP2, we get
l?
M
1.56 GeV.
(8.14)
If QCD correction is taken into account, then
r M 1.43GeV which gives the life time of
T
(8.15)
to be
= 4.60 x
(8.16)
Thus t quark decays before it can form bound states such as tf and
tq.
Heavy Baryons
267
8.3 Heavy Baryons Since u, d, s belong to the triplet representation of SU(3), the charmed and bottom baryons with spin parity belong to either triplet representation 5 or sextet representation 6 of SU(3). Using the Pauli principle, the unitary spin and spin wave functions of spin baryons can be written as
;+
4'
(8.17) (8.18) where i, j = 1, 2, 3 (ql = u, q 2 = d , q 3 = s, Q = c or b ) and the spin wave functions X M A and X M S are given in Eq.(6.8) and Table (6.3) respectively. Note that A i j belongs to triplet representation 3 of SU(3). In particular we have an isospin singlet and isospin doublet: A12 = A!(A:),
A13
-+
-0
-0
--)
= -Ec (Zb), A 2 3 = =,(=b
(8.19)
Sij belongs to the sextet representation of SU(3). In particular, we have an isospin triplet, an isospin doublet and an isospin singlet: s11
=
s12
=
s22
=
s13
=
s23
= =
s33
;+
++ (c+ b >,
c,'(m, Ac:(c,), n'o EC(Eb 1 7 n'+
&O -c
n'-
(=b
An:(n,).
(8.20)
The spin baryons also belong to the sextet representation of SU(3). They are given by Eq.(18), with X M S replaced by xs where the spin wave functions xs are given in Table (6.3). The six baryons are labelled as spin
;+
Heavy Flavors
268
In addition to C = +1 and B = -1 baryons considered above, we also have the following baryons with C = 2 and B = -2 belonging to the triplet representation of SU(3) with spin parity (3/2)+:
Finally we have singlets with C = 3 and B = -3, namely
Experimentally only one bottom baryon has been detected so far. Its mass and life time are V L A ~ = 5641 f 50 MeV and T = (1.14 f 0.08) x 1 0 - l ' ~ . Some of the charmed baryons have been discovered experimentally, they are given in Table 3. 8.4
Quarkonium
The bound system of heavy quarks QQ, Q = c, b, is called quarkonium e.g. charmonium cC, bottomonium bb, Since quarks are fermions with spin 1/2, their bound system can be written as ( Q Q ) L , s . Now S can have two values 0 and 1 with spin wave function antisymmetric and symmetric respectively. If we regard Q and Q as identical fermions which differ only in their charges, then we can state generalized Pauli principle: The wave function is antisymmetric with the exchange of particles Q and Q. Under particle exchange, we get with space coordinates exchange, a factor (-l)",with spin coordinates exchange, a factor (-l)s+l and with charge exchange, a factor C (C is called C-parity). Hence Pauli principle gives
269
Quarkonium
Table 8.3
270
Heavy Flavors
Therefore,
c = (-1y+S.
(8.24)
Hence we have the result
-1 L + S C = { +1 L + S
odd even.
(8.25)
Also for (QQ) system, the parity
P = (-l)(-l)L
= (-1)L+1
(8.26)
Let us now use the spectroscopic notation,
L =
0,
1, 2,
3,
S, P, D, El
* * . a
*
*
I
A state is completely specified as
where n is the principal quantum number and J is the total angular momentum. Thus for L = 0, we have the following states n 'So C = + 1 , n = 1 , 2 , . . . n 'So C = -1, n = 1, 2,
,
The ground state is therefore a hyperfine doublet 1 'So(O-+) and 13S1(1--).For L = 1, we have the following states
n lPJ J = + 1 , C=-1, n 3 P ~J = 0, 1, 2 C = 1, Finally, we note that for L n TI,
= 2,
1+Of+, I++, 2++.
we have the following states
'DJ J = 2 , C = + l , 2-+ 3 D J J = 1, 2, 3 ( ? = - I , I--, 2--, 3--.
Quar konium
271
It is interesting to see that the state 3D1 has the same quantum number as 3S1.They can therefore mix, but the mixing is expected to be small. The states 3PJ and ‘PI is a hyperfine quartet (degenerate), but this degeneracy is removed due to hyperfine splitting. The low lying states listed above are shown in Figs. 3 and 4. Most of these states have been discovered experimentally and they are listed in Tables 4 and 5. The transitions and decays of charmonium states are shown in Fig. 5. Similar transitions and decays occur for bottomonium bound states. From Fig. 5, we note that both A41 and El radiative transitions are possible: J/Q
+
qc+y 7, y M1 transitions 7: y (no parity change)
+
Q+y
-+
x +y
+
qc
+
9’ + Q’
9‘
x
+ +
El transitions
+ y (parity changes).
From Eq.(6.87) and Table (6.7), we get (for example) r(QI
+
VCY)
[-I
4 a 2 1 k3R 3 3m, = 2.7 keVR =
(8.27)
where R is the overlap integral defined as
and q = (m,,/m)k, k is the momentum carried by photon, mspis the mass of the spectator quark and m is the mass of the bound state. For R = 1, r is about a factor of three larger than the experimental value.
Heavy Flavors
272
-lDz _ _ _ _ _ _ . C h a r m threshold
Figure 3
The charmonium spectrum ( cE bound state).
Table 8.4 The
11, family
(cc) bound states.
Qumkonium
273
tys 0
0-+
1--
1+Figure 4
(0 I 2 ) j (1 2 3j-
274
Heavy Flavors
Table 8.5 The 'Y family (bb) bound states.
275
Quaxkonium
r
0-+
MI El
1--
1+- (0, 1,2>*
(1,2,3)-- 2-+
Figure 5 The transitions and decays of charmonium states.
For El transitions nS1 -+ n’PJ and nPJ -+ n’S1 ( J = 0, 1, 2) the decay widths can be written (cf. Eq.(6.68)) (8.29) (8.30) where
Mntn
=
x
< Q > antn,On), = (l/h)
Jom Lo(P)
- 2j2 (QT)]%o(r) &I
(r)r3dr. (8.31)
Note that j o and j , are spherical Bessel functions and R,l are radial wave functions. In order to predict these decay widths one needs to know the radial wave functions, i.e. some potential model is needed. Finally, we note that there are 22 states below B threshold as compared with eight states below charm threshold. This is a
Heavy Flavors
276
Figure 6 consequence of the fact that interquark potential is flavor independent (as expected in QCD) so that En, - Enl is the same for cC and bb. (Note that charm threshold is at about 3.74 GeV whereas B threshold is at about 10.55 GeV.) 8.5
Leptonic Decay Width of Quarkonium
The decays of 'S1(&Q)state ( V ) into charged leptons proceeds through the virtual photon as shown in Fig. 6. The scattering cross section for the QQ -+ It is given by Eq.(A.78) u =
471-cY2 1 Pl -(&>, -3 PQ
where
(8.33)
277
Leptonic Decay Width of Quarkonium
and Q is the charge of the quark Q. Now the cross section u can be written as
(8.34) where ot is the cross section for 3S1state and usis the cross-section for IS0 state. Since the photon is coupled to a conserved vector current, therefore it contributes only to spin triplet state, Thus os= 0. Hence the decay rate in the limit pl 1 (s = 4772; >> 47723 is given by --f
F
=
(incident flux) ct
=
2pQIq 8 (0) I2
4
(8.35)
where the incident flux = p i , ( 2 , 8 ~ ) = 2(lk8(o)(2,8Q. Hence from Eqs. (32) and (34), we get (8.36)
where we have put s = 4m; M m t and PQ M 0 (in the nonrelativistic limit). Taking into account the color IV >= $ C, >, we multiply Eq.(35) by a factor of three. Hence we have
(8.37) It may be pointed out that, before comparing experimental leptonic widths with their theoretical predictions, the vacuum polarization contributions to the leptonic decay width have to be removed so that r0 = reyl
- n)2
where (1 - 11)2= 0.958, 0.932 for charmonium and bottomonium respectively and then it is r0 which is to be compared with the theoretical predictions.
Heavy Flavors
278
Figure 7
8.6
Positronium
(lS0
state) decay into two photons.
Hadronic Decay Width
The decays of quarkonium states 3S1and 'So to ordinary hadrons are suppressed by the 021 rule. The narrowness of their decay widths can be explained as follows. By C-conservation 3S1state can decay in the lowest order to three gluons and thus its hadronic decay width is proportional to a: x (probability of conversion of gluons into hadrons). Since color is confined so this probability is unity. Similarly the decay of 'So into hadrons is proportional to a:, since by C-conservation it can decay into two gluons. Here analogy with positronium is in order. Positronium in 'SOstate (para positronium) decay into two photons via the diagram (Fig.
7). In the low energy limit the cross section for the above process is given by 2
(8.38)
a=;(;).
Since at = 0, we get using Eq.(34) u s
=4a=
47r
(--)
(2
.
(8.39J
Hence the decay rate
r [ls,,(e-e+)
a2 271 = IpXPs(0)124a= 1 6 ~ ~ ~ Q ~(8.40) ( 0 ) ~ ~ 4%
279
Hadronic Decay Width
For (QQ) 'So state decaying into 27, we replace e4 --t
and
4771:
-+
4,;
M
[hQ 2 e2 ] 2 = 3Q4e4
m;. Hence we get
I' ['So(mp)+ 273
(8.41)
=
For qc 2 gluons, we replace a2 by $a: in Eq. (40) [see problem 21, so that we get the hadronic decay rate ---f
The decay rate for 3Sl(e-e+) system going to 37 is given by
For the decay of 3S1(QQ) --t 39, we replace a3 by 5a:/18 [see problem 21 and (2me)2= ( 2 m ~M) m$ ~ in Eq.(43). Hence we get
r [3S,(V)
-+
hadrons] = F [3S1-+ 3g] -
160n(r2 - 9) a3 81n m$ I QS (0) I2 . (8.44)
We now apply the above results to From Eqs.(44) and (37), we get
4, J / @ and Y decays.
From Eq.(45), we get as(md)M 0.44,
as(m*) M 0.22,
a s ( m ~M)0.18,
Heavy Flavors
280
where we have used r(q5 + non-strange mesons) x 653 keV, l?(J/Q--f hadrons) M 76.5 keV, r ( T -t hadrons) x 50 keV, l?($ + e+e-) M 1.37 keV, I'(J/e + e+e-) x 5.26 keV, r(T e+e-) M 1.32 keV. From this we see a realization of the asymptotic freedom of QCD, the coupling a,(q2)falls with the increase of q2. Finally from Eqs.(42), (44) and (37), we have [with aLIS(mqc) = --f
Qs
I)
(ma
r(qC --+ hadrons) =
271~ mi 1 I? ( J / Q + hadrons) 5(r2- 9) mic Q,(md (8.46)
x 7.6MeV
(8.47)
where we have used a,(m*) % 0.22. This value is lower than the experimental value rtotx 13.2:;:; MeV for qc.
Non-Relativistic Treatment of Quarkonium
8.7
From a theoretical point of view, heavy quark system (quarkonium) is interesting because this is a relatively simple system. To a good approximation, the quark motion in this bound state should be non-relativistic. Thus we can use the Schrodinger equation for QQ systern :
ti2 (8.48) --V2Q(r) [ V ( r )- E ] Q ( ~ =)o 21-L p is the reduced mass of QQ system i.e. p = imo. For central potential, we can use the wave function:
+
$1.
(8.49)
Q(r) = ~ ( r ) Y 1 m ( 4
The radial wave function R ( r ) satisfies the equation
[-
h2 d2
--
2p
+ dr2
R ( r )-
I. v(r) -
-
]
1(1+ l)h2 R ( r ) = 0. 2pr2 (8.50)
281
Non-Relativistic Treatment of Quarkonium
If we define a radial function (8.51) then x ( r ) satisfies the equation
+
"I
- V ( r ) )- r2
= O.
(8.52)
The wave function x ( r ) is normalized as
= I
(8.53)
with the boundary conditions x ( 0 ) = 0.
(8.54)
For S-waves: (8.55) For S-waves:
xyo) = R(0)= &QS(O)
(8.56)
We now prove two important results:
1. (8.57)
Proof. From Eq. (52) for I
=0
(8.58)
Heavy Flavors
282
Therefore, (8.59) Taking the expectation value [note X(r) is real], we get dV
00
(8.60)
Integrating left, hand side by parts, we get
=
[x’(O>l2 = [R(0)I2,
(8.61)
where we have used the boundary conditions (54) - (56). Hence Eq. (60) gives
2. Virial Theorem
:(2 )
( T ) = - r-
.
(8.62)
Proof. From Eq. (59), we have (8.63) Integrating left hand side by parts and using Eqs. (54) and ( 5 5 ) , we get 00
1.h.s. = 2 1 Xl’dr.
(8.64)
283
Non-Relativistic Treatment of Quarkonium
Therefore, (8.65) Now from Eq. (58),
(Z)
=
-$ [E- < v >].
(8.66)
But
($)
= Jdm&'dr =
1
00
I-xx'l;
-
XI2dr
= -1m~'2dr.
(8.67)
Hence from Eqs. (65) - (67), we get
E - (v)= ( rdVz ) or
( T )=
;
(8.68)
dV
( I T )
*
Let us apply Eq. (62) to one gluon exchange potential V ( r )= -$a S 1. T For this case
(f)
=
;as
(;)
= -as-, 2 1
3
a
(8.69)
where a = 3/4pas is the Bohr radius. Thus, we get _w -- -2a s . c 3
(8.70)
Heavy Flavors
284
As a, decreases with mass, for sufficiently high mass v / c << 1 and one can treat dynamics non-relativistically. For the special case of power law potential
V(T= ) A + AT’,
(8.71)
one can obtain interesting results by studying the scaling of Schrodinger equation (52). Put p = ,Or, where p is some parameter such that it makes p dimensionless. Let us put X ( T ) = u ( p ) and I? = E - A. Then we get from Eq.(52)
Put
1x1 = $,02+u,this gives (8.73)
Then we put (8.74) where E is dimensionless. If we write sgn(X) = X/lXl, we obtain Eq.(71) as E 7 sgn(X)p”
-
(8.75)
which depends only upon pure numbers. We now study the consequences of Eq. (75). (i) Lenghts and quantities with the dimensions of length depend upon constituent quark mass rn = 2p and coupling strength 1x1 as (8.76)
Non-Relativistic Treatment of Quarkonium
285
Simple Coulomb harmonic like oscillator linear u=-1 v=2 v=l 4
P3/ P-lI2
P
log v=o P3/2
7 constant
Power law v=o.1
P1.43 p-0.048
Particle density at the origin of coordinates IQS(O)l2
L-3 cX (PIXI)- 1 / 2 w .
(ii) Level spacing between energy levels depends on p and
AE
-(plA1)2/2+u 1
P-'/2+vJx12/2+~
P
(8.77)
1x1 as (8.78)
The "power law" potential corresponding to the limiting value v 0 is simply the logarithmic potential.
r
V(T= ) Cln--. r0
---f
(8.79)
We summarize these results for the power law potentials in Table 6.
Observations
rnr)- rnr
= rnqt
- rnq
(8.80)
implies either u = 0 or u is very small. In fact Martin has shown that the potential
V(r) = -8.04 GeV
+ 6.87O(r/l GeV-')'.'
gives a good fit to quarkonium mass spectrum.
(8.81)
Heavy Flavors
286
The logarithmic potential
V(T= ) (0.71 GeV) In
(8.82)
also gives good fit to the data. The two forms are numerically indistinguishable for 0.1 fm5 r 5 1fm. If we plot 1q3(0)l2for the vector bosons p, w , $, Q, Y versus p in a log - log plot, a straight line fit is possible i.e.
1.6. Again this supports the power law potential with v with p very small, i.e., v 0.1. It is clear from Tables 4 and 5 that both for charmonium and bottomonium, the low lying bound state energy spectrum satisfies the rule N
N
El3
< E23*
(8.84)
In particular we find for cC
fit(13P)- m ( l S ) = m(z3S1)- m(1 3S1) = m ( l 3S1)- m(1 'So) = ~ n ( 2 ~ S-1 m) ( 2 ' S O ) =
457MeV
q' - J / q = 589MeV J / Q - qC = 117MeV @'
(8.85)
- 7; = 82 MeV,
where
m(S)
=
m(3P)
=
+ +
3m(3S1) m(lS0) 4 5m(3P2) ~ v L ( ~ P om(3Po) ) 9
+
(8.86) (8.87)
For Coulomb potential, the energy spectrum satisfies the rule El8 < E23 = €3,
< E3s = E3p = E 3 d
(8.88)
Problems
287
and for the harmonic oscillator potential
Further, the harmonic oscillator potential gives the level spacing as follows: 1 (8.90) Eip - El, = E23 - Eip = Z(E28 - Eis). Thus although oscillator potential is a confining potential, the level spacing is not in agreement with the experimental results. The QCD inspired Cornell potential [cf. Eq. (7.62a)I
K r V ( r )= c - - + r a2 reproduces the mass spectrum for CC and bb bound states quite well (see problem 3). Thus we see that the quarkonium spectroscopy is consistent with a potential that increases linearly at large distances, thereby supporting the color confinement. We also saw in this chapter (as well as in Chap. 7) a realization of other striking property of QCD, namely the running of the QCD coupling constant a3(q2) with q 2 . 8.8
Problems
1. Show that if J / 9 has isospin I = 0,
Hint: The p r final state has I = 0, 1 or 2. But we are interested in I = 0. Using C. G. coefficients (p-7T+)
=
1 3 l0,O) +
* *
1
d3 l0,O) +
lPOTO) = --
*
Heavy Flavors
288
Figure 8
2. For (4Q)color singlet + 27 or 2 gluons, as shown in Fig. 8, where a, b = 1 , 2 , 3 are 3 colors of quarks, A , B = 1, , 8 are eight colors of gluons, show that
-
M(2.9) -=--
a3
W ~ Y )aQ:
9
i'AB
3
and hence show that
For (QQ)colorsinglet, 3 37 or 3 gluons coupled symmetrically in gluon color, show that
5
Hint: Use A d ~ s c d ~ = Bc 3. By writing
E = < H >=< q
H p >= 2 m +
1 - < p2 > 2P
+
+ < V ( r )>
where p = m/2 and V ( T )= C $7, evaluate the energy eigenvalues El,, El, and E2s by variational principle.
Problems
289
Hint: Write E = (2pb4)'l3E = (2pb4)'I3[< H > -2m ?!, + C) and @ = $Km(6', q5) and express
z= S[-us
- C ] ,( E =
+ (y - + y) u2] dy s U2dY
where y = (2p/b2)1/3rand 7 = (4p2b2)'l3K.[All these quantities are dimensionless and are therefore also suitable for numerical solution on a computer.] Using the trial wave functions 1s ; = Nye-1/2p2Y2
1P : u = N y 2e -1/2p2y2
,
minimize E in order to determine the parameter for each wave function. Then find Z. For numerical purpose, use m, = 1.52 GeV, K = 0.48, a = 2.34 GeV-l. Compare your results with the experiment a1 values. Using the equation 16ns(0)12= lL 2~
(dr5 ) = .E[' + K (f) ns] , 2~ a2 12s
Vector Meson mv (MeV) 770 P W 783 1020 4 6 3097
l? (keV) 6.77 i0.32 0.60 f 0.02 1.37 f 0.05 4.72 f 0.35
290
Heavy Flavors
find ~ @ n s ( 0 ) ~ ~and , u ,using ~ , q the values thus found and the dependence of I\k,,(0)12 on p = m,/2 discussed in Table 6 for various potentials, find which potential(s) are favored.
Bibliography 8.9
291
Bibliography 1. F.E. Close, An Introduction to Quarks and Patrons, Academic Press, London (1997); Reports on Progress in Physics, 51, 583 (1989). 2. C. Quigg and J. L. Rosner, Phys. Rep. 56, 167 (1979). 3. H. Grosse and A. Martin, Phys. Rep. 60, 341 (1980). 4. R. N. Cahn (editor), e+e- Annihilation: New Quarks and Leptons (Annual Reviews Special Collections Program), Benjamin/Cummings, Menlo Park, California, 1985. 5. E. Eichten, "The Last Hurrah for Quarkonium Physics: The Top System", in The Sixth Quark, Proceedings of the 1984 SLAC Summer Institute in Particle Physics, edited by Patricia M. McDonough, Stanford Linear Accelerator Center Report SLAC-281, January, 1985, p. 1. 6. M. E. Peskin, "Aspects of the Dynamics of Heavy Quark Systems", in Dynamics and Spectroscopy at High Energy, Proceedings of the 1983 SLAC Summer Institute in Particle Physics, edited by Patricia M. McDonough, Stanford Linear Accelerator Report SLAC-267, p. 151. 7. C. Quigg, Quantum Chromodynamics near the Confinement Limit, Fermi Lab.-Conf.-85/126-T (1985). 8. J. Lee-Franzini, Nucl. Phys. B3, 139 (1988). 9. High Energy Electron-Positron Physics, Eds. A. Ali and P. Soding, World Scientific, Singapore (1988). 10. Particle Data Group, Z. Phys. C3, 1-4 (1998). 11. J.L. Rosner, Heavy Quarks, Quark Mixing and C. P. Violations, in testing the Standard Model [TASl-gO] Editors M. Cvetic and P. Langacker, World Scientific, Singapore, 1991. 12. W.Lucha, F.F.Schober1 and D.Gromes, Phys. Rep. 200, 127 (1991). 13. H. Grosse and A. Martin, Particle Physics and Schrodinger Equation, Cambridge University Press (1999).
Chapter 9 HEAVY QUARK EFFECTIVE THEORY (HQET) 9.1
Effective Lagrangian
The QCD Lagrangian (7.32) for light quarks is chiral invariant in --+ 0. For a heavy quark c, b, or t , the chiral symmethe limit try does not hold. However, QCD has asymptotic freedom which implies that the effective coupling constant asdecreases logarithmically at short distances or high momentum transfers. This is the basis for perturbative QCD i.e. above a certain mass scale p, the perturbative QCD is applicable. The size of a hadron is of the order of AQCD, where AQcD 0.2 GeV (see Chap. 7). Thus for a bound state of quarks or (quark-antiquark), we are in the nonperturbative regime i.e. in the confinement region. Consider for example a bound state of light-heavy quarkantiquark, viz. qQ or Qij. In the limit m~ --t 00, heavy quark (anti quark) can be taken as a static source of field in which light antiquark (quark) moves. The situation is like hydrogen atom. In the limit WIQ -+ 00, the Hamiltonian for the light degree of freedom in analogy with H atom can be written to order u2/c2 N
+ K(T) - p^4 - 1 2mq 8mi 4mi
H = - P2
-C
1 8mi
.(E" x 0 ) - -7
*
EC,(9.1)
where E"is the color electric field, K ( r ) is related to Ecby E" = dV r ---$;, and @ = - 2 ~ . Although X Q ~ D / ~isQsmall, but it is still finite. The effective heavy quark theory provides a framework to take into account l/mQ corrections. 293
Heavy Quark Effective Theory (HQET)
294
The starting point is to define a four velocit,y
vo = y = uo so that 7)
2
=
‘fJp7)p
= y2 - y2U2 = y2(1 - U2) =
1.
(9.3)
It is convenient, to define
.JI = YILV/L,
.JI2 = ? I 2 =
1.
(9.4)
The Dirac equation for a heavy quark is given by (iy’”D, - m)Q = 0
(9.5)
where D, is the covariant, derivative
G, = X,4/2G; We define the projection operators
P*
1
=
2 (1f $) .
Note that in the rest frame v = 0 1
P* = 2(1
* yo)
i.e. it, projects out, upper and lower components of 8 . Write
(9.7)
295
Effective Lagrangian
P-yc" = yPP+
-up.
(9.10)
Using Eqs. (8)-(lo), we obtain from Eq. ( 5 ) (iy. D
+ iv
*
D)h-,
+ iv
*
Dh+, = 0
+
-
(iy D - iv . D)h+, - (2m iv D)h-, = 0.
(9.11) (9.12)
Note that h+, and h-, are not decoupled. We show that to order l/m2, the equations for h,, and h-, are decoupled. From Eq. (12), we obtain iy. D - i~ D 2 m + i v . D h+U 1 iv D = -[1-++.*][iy*D-i~*D]h+,. 2m 2m *
h-,
=
9
(9.13)
Thus from Eqs. (11) and (13) to order l/m, we get [iy * D
- iv . D + iv - D ]iy . D2m h+v + iv - Dh+, = 0.
(9.14)
Now
=
s ~"G~,. D2 - -9 O 2
Hence from Eq. (14), we obtain [ZV
*
D2 9s D -+ -gPvGPV 2m
4m
(iv 0)' ]h+,= 0. 2m
(9.16)
Heavy Quark Effective Theory (HQET)
296
This is the Pauli form of Dirac equat,ion t,o order l/m. The corresponding Lagrangian for the field h+, is given by
Note that h,, annihilates a heavy quark. In the limit m -+
C,ff
=
00
h+,i?)* Dh+,.
(9.17b)
[P,, Q(z)]
(9.18)
Now from the relation
-28,Q(Z)
=
it follows through the transformation (8) that
-ia,h+,
= mu,h+,
+ [pp,h+,].
(9.19)
This shows that, a derivative acting on h+, corresponds t80a factor of the residual moment,um k, carried by t,he heavy quark
- k , = mup - p ,
(9.20)
so that, k, indicates how milch heavy quark is offmass shell. In the limit m + 00 (no recoil limit) with 71, and k , fixed
(9.21) One would expect the heavy quark to carry most of the momentum of the QQ bound state, but not all: p PB = p ,
+ 1,
=m
u r k + 1,
(9.22)
where p,” = rrlgiip is the momentum of the bound system and 1, is that, which is carried by the light, degree of freedom. Now m B = m + mq - B where B is the binding energy supplied by the interaction through gliion. Thus from Eq. (22) (9.23)
297
Effective Lagrangian
so that again v r k + U , as m --t 00 and a comparison of Eq.(23) with Eq.(21) shows that in the limit m --t 00, the interaction with gluons can change k, but not v r b = v,; the velocity of the heavy quark can be altered only by an external current which absorbs ”infinite momentum”. Thus in a hadron, the light degrees of freedom are independent of the heavy quark mass i.e. residual motion of the heavy quark in a hadron can be taken into account by adding the effective Hamiltonian for heavy quark Q from the L,ff given in Eq. (17) to the Hamiltonian for the light quark given in Eq. (I). We will come to this point later, when we discuss the masses of heavy hadrons. The third term in the Lagrangian (17a) undergoes shortdistance QCD corrections and as such is multiplied by the renormalization factor (9.24) with
ZQ ( p = m) = 1.
The heavy quark propagator in QCD can be written in HQET using Eq. (20): (9.25) where we have neglected the term k 2 / m -+ 0. The gluon heavy quark vertex can be written from the Lagrangian (17) and is given bY ig,vl” (Ts),“. (9.26) The following relations are useful $7, +7,$
= 271’
#rp# =
2v,$-r,.
(9.27)
From Eq. (27), one can write, on using Eq. (9),
h+vYph+v = h+vvph+,.
(9.28)
Heavy Quark Effective Theory (HQET)
298
9.2
Spin Symmetry of Heavy Quark
In the limit m --f 00, the Lagrangian L e f f given in Eq.(17) has additional symmetries not present in the full QCD Lagrangian. One such symmetry namely the spin symmetry of heavy quark is reflected in the fact that the first, term in t,he Lagrangian (17) viz. Eq.(l7b) makes no reference to the Dirac striict,iire at, all which can couple to the spin degrees of h+,. More explicitly define t,he spin: si = -s; = -yS#y
e,
(9.29)
where
(9.30) In the rest frame of h,
Thus in the rest, frame
(9.31) i.e. we get the usual definition of the spin. We note that the Lagrangian L e f fgiven in Eq.(l7a) is invariant under the infinitesimal transformation 6h+, =
ie.sh+,
~ h + ,= -iO. sh+,.
(9.32)
Now the Noether current is given by
(9.33)
Spin Symmetry of Heavy Quark
299
Hence the spin operator is given by
S
=
/Jo(x,t)d3z
= vo
We note that
pi,h,,]
1
h+,sh+,d3x.
= -Sib+,.
(9.34) (9.35)
We conclude that the Lagrangian L,.. in Eq.( 17a) is invariant under SU(2) of heavy quark spin symmetry. It means that pseudoscalar and vector meson states IP (w))and IV (w,e ) ) ,containing same heavy quark, with momentum p,” = mgwP can be related to each other: Ss (u)JP(w)) = -i J V(w,E ) )
, S3 (w)J P(w))= i J V(w,e)) . (9.36)
Thus their masses are degenerate in this limit. This degeneracy is lifted by the third term in the Lagrangian (17) giving e.g. for B* (1-) and B(0-) mesons, the mass difference (mk - mg) which scales like l/mb. The second symmetry of the Lagrangian (17) in the limit m + 00 arises when we introduce two distinct flavors hl (e.g.b)and hZ(e.g.c). Since the first term in Eq.(17) makes no reference to masses mi (i = 1 , 2 ) , and since mass is the only property which can distinguish between quarks of different flavors in QCD, the effective theory has a symmetry under which hl ( u ) ++ hz ( u ) . It may be emphasized that this symmetry does not in any way depend on ml = ma but only on mi >> A, where A is a scale parameter such that l/h determines the size of light degrees of freedom in the bound state and is a few hundred M e V ; it may vary from process to process. Note also that the flavor symmetry holds between heavy quark fields of same velocity and not with the same momentum. This flavor symmetry together with the spin symmetry mentioned above gives rise to SU(4) symmetry for the h1).( and has been used to relate the matrix elements system hz (v)
(
)
300
Heavy Quark Effective Theory (HQET)
of flavor changing effective currents which mediate weak decays of mesons containing heavy quarks. Since we will be dealing with h+, only, we will drop the subscript in what follows. We first note that
+
These equations follow from Eq.(35). Their use is as follows: Consider the transition B- (w)+ D (w'). Then from Eq.(37), we obtain
I bi,E , x ~ , ]
( D O(w'>
I
I B - (v)) = ( D O(w'> [q,,Sirb,] I B - (.,>
.
(9.38) Now using Eq. (36), we get
we get
(9.42)
Spin Symmetry of Heavy Quark
where we have used the fact that since E so that
30 1 ~ p is b
a symmetry current,
with
t o (w2) = t o (1) = 1.
(9.44)
Another application of Eq. (37) is for the matrix elements of the current q7” (1 - 75)b ( q = u , d , s) between the vacuum and B meson state viz.
(01
[S;, qrb] 113,) = - (01 qrsibp4) .
(9.45)
Hence we get
i (01 Q r b Iq)= - (01 qr (75$Y
*
&)
b IB,)
*
(9.46)
Thus for I? = yp (1 - y5) on using Eqs. (40) and (41), we obtain
where we have used
and
The results obtained in Eqs. (42) and (47) will be used in Chap. 16, where we will discuss the semileptonic decays of B mesons involving the vector and axial form factors in the transitions B t D , D‘.
Heavy Quark Effective Theory (HQET)
302
Similar results can also be derived by the following procedure (called trace technique). For vector and pseudoscalar meson fields, we can write ( P = B or D ) (9.50) where a = 1 , 2 , 3for u,d and s quarks and P& and Pa are annihilat,ion operators normalized as
(0 (palQ
Tia
(0-))
=
1.
(9.52)
We define the adjoint field (9.53) We note that
(9.54) The spin symmetry which relates Pa and P:p is automatically incorporated in Eq. (50). In case of spinor field q ( x ) ,the wave function can be written
Then in view of Eqs. (50)-(52),we can write for mesons containing a heavy quark
l+d
(0 [ H a l Pa (u)) = -2
T5
(9.56)
(9.57)
Mass Spectroscopy for Hadrons with One Heavy Quark
303
We now apply the above considerations for the matrix elements (O-(V’)IJAIO-(U)) and (l-(w’,e*) I J x l O - ( w ) ) whereJx = Vx-Ax. Using,Eqs. (50)-(53), (56) and (57), we get
(9.58)
(9.59) Note that since only vector current contributes in Eq.(58), the form factor (-u- d)is normalized as 5 (1) = 1. Comparing Eqs. (58) and (59) with Eqs. (40) and (46), we see that trace technique gives exactly .the same results for B --t D(D*) transitions as previously obtained. 9.3 Mass Spectroscopy for Hadrons with One Heavy Quark
We now discuss the effective Lagrangian (17a) in relation to hadronic masses containing one heavy quark. We introduce the following notation: Write a pseudoscalar (vector) heavy meson as P,(P,*), P = B or D ,4 = u, d or s and we take mu = md. The heavy baryon is written as BQ, Q = b or c, B=A,E,E,Z’ or il ; the light quark content and spin configurations are contained in these symbols.
Heavy Quark Effective Theory (HQET)
304
Define -
(9.60a)
iz ( P )= (PI &D2h, IP)
d ( P )= ZQ ( p ) (PI g ~ L ~ ’ ” G( P~).~ h ~ (9.60b)
We take iz(P) and z ( P ) independent of the light, quark flavor q . We also assume iz(B)=E(D)=8 (9.6 1a) (9.6 1b) These assumptions imply that interquark interactions are flavor independent. To understand the physical meaning of these terms we go to the rest frame of Q,v = 0. In this frame iv
*
D
= g8G,
(9.62)
- D 2= (V- ig,G) (V- i.g,G) = D2 =
(
i n , EC
0
where G4j =
-Ej” , Bca
+2
( “iBc 1
= -2& .v. k G~k .
u
*
B“
(9.63)
)
(9.64a) (9.6413)
are the colour electric and magnetic fields respectively. Thus &,D2h, is gauge invariant extension of kinetic energy term representing the residual motion of heavy quark in a hadron and term h,cQ.BChu describes the color magnetic coupling of the heavy-quark spin to the gluon field (S = ;“)[we have exhibited the subscript Q with 61.
Matching Eq. (62) with V, in Eq. (l),we can write the effective Hamiltonian for a bound hadron containing one heavy quark Q as H=H,+HQ (9.65) where HQ takes care of the residual motion of the heavy quark and in view of Eqs.(63) and (64), it is obtained from C,ff in Eq. (17a) to order l / m ~as follows:
Mass Spectroscopy for Hadrons with One Heavy Quark
305
(9.66) Note that the second term on the right hand side of Eq.(66) represents the interaction of color magnetic field B"with color magnetic (ZQ( p )g 8 ) . The Hamiltomoment of the heavy quark pQ = nian (65) gives the mass of the heavy meson P, as h d(P) mpq = mQ+ A , -- (9.67) 2mQ 2mQ
+
where we have put
A,
+ m,.
(9.68)
= (Hq)
gives rise To proceed further, we note that the term OQ*B" to color magnetic moment interaction of the type pq.pQwhich is is )-3 or 1 for proportional to O,-UQ and as is well known ( ~ , . u Q spin singlet ('So) or spin triplet (3S1)states respectively. Hence relative to Eq.(67) for the heavy meson P:, we obtain (9.69) We obtain from Eqs. (67) and (69) the mass relations
'
mD; - mDd - mDs*-mo, = -- (Dl 3 mc
(9.70b)
Heavy Quark Effective Theory (HQET)
306
where m p =
+m p
3mp* 4
(9.72)
Experimentally (in MeV) mDL; - m D d m B ; - mBd
m B , - mBd
mB
142.12 f 0.07, m D ; - m D , = 143.8 f 0.4 = 45.7 f 0.4, m ~ -;m ~=, 47.0 f 2.6 = 90.2 f 2.2, m D ; - mDd = 99.2 f 0.5 .= 5.313GeV, W ~ = D 1.971 GeV. ==
From Eqs.(70) and (61b), we also obtain
Using the experimental values for the masses, we obt,ain as (mb) as
(m,)
0.69
(9.73b)
Eq. (74b) is compatible with a, (mb) N 0.22, as (m,) N 0.32 [see Chap. 161 used in discussing the decays of D and B mesons. If we use mb = 4.9, m, = 1.5 (in GeV), then from Eq. (70), we get d ( B ) - 0.07 GeV, z ( B ) N 0.34 GeV2 (9.74a) -
N
mb
-
(D) N
mc
-
0.213 GeV,
2( D ) N 0.32 GeV’.
(9.7413)
It may be noted that we cannot use Eq.(72) to determine si, in a meaningful way since it, is very sensitive to quark masses mb , m, which are not, well known. Finally we note from Eqs.(67), (69) and (73) that, m B
- mb =
+0
(9.75a)
307
Mass Spectroscopy for Hadrons with One Heavy Quark
which gives for mb N 4.9 GeV,
k,
11 0.41
GeV.
(9.75b)
With the help of Eqs. (7.91) and (7.92), we can derive the mass formulae for heavy baryons using Eqs (65) and (66). We obtain for the baryons AQ , CQ and Eb, the masses (9.76)
where ii and ;are given in Eqs. (60) with P replaced by BQ and Ad =
(H,) -k md
+ mu
(9.78)
and parameter arises from the color magnetic moment interaction of light quarks in a heavy baryon[cf. Eq. (7.66)]: (9.79) €+om Eq.(80), one can write
(9.80a) where (9.80b)
The masses for other baryons can be obtained from mAq , mz, and mc; by appropriate replacement in the light flavor index. F'rom Eqs. (77) and (78), we get I
m p - mcQ= mz;, - mEQ= mnb - mnQ- 3z(CQ) Q
2mQ
(9.81)
308
Heavy Quark Effective Theory (HQET)
1 -2 ( m , Q + mQ,
)-
mSQ '
12 (mx; + m,,
=
Q
'
( m -~m ~~ = ~(mb)- mc)
].
(9.83)
The present experimental value for the charmed baryons are given in Table (8.3). From this table we find
mCf
- mEc- m,, -c
- m'=c
N
65 MeV.
(9.84)
Thus the mass relation ( 8 2 ) is well satisfied. From Eq. ( 8 2 ) then we obtain m": = m n c 65 MeV N 2765 MeV. (9.85)
+
We also note from Eqs. (75) , ( 8 2 ) and (85) that, (9.86) Further from Eqs.(72) and (84), we get
Now mAb- mAc21 3.34 GeVz m B - mD,therefore 6 = a. Using E =? wei, get from Eqs. (71), (72), (77), and (78): mAh- m, = mAc- m,
-
where
-
1
-m"Q- 4
-
_
= Ad - Ad
[m +mCQ+2rn,, "0
Thus using mAc = 2.443 GeV,we get - & = 0.47 GeV.
Q
1
.
(9.88) (9.89)
(9.90)
Also from Eqs. (77) and (78) (9.91)
The P-wave Heavy Mesons: Mass Spectroscopy
309
Thus in particular for charmed baryons, we have - 5-
+ 32
-
(m&- m&.)
N
168 + 43 N 211 MeV
mumd
(m*,c
- m& (9.92)
which is not very much different from N 213 MeV (see Eq. 75) . The success of the mass formulae Eqs. (70) and (82) cannot be taken as verification of HQET, since similar formulae also hold for light hadrons (see Eqs. 7.80, 7.81, 7.82, 7.98, 7.99). If we follow the approach of Chap. 7 for baryons and put, (9.93) then Eq. (83) remains unchanged. Using mEc- mAc= 168 MeV, we obtain
N
67 MeV 20 MeV
(9.94)
N
199MeV
(9.95)
m,,= - mEc
N
mxi - mxb mXb
- mA,
i.e. the results similar to the ones obtained in HQET
9.4 The P-wave Heavy Mesons: Mass Spectroscopy
So far we have discussed only S-wave heavy mesons. We now discuss P-wave mesons for which experimental evidence is available. Since the spin of heavy quark is decoupled, it is natural to couple orbital angular momentum L with S, in the heavy quark limit. Thus we define (9.96) j = L S,.
+
Now the total angular momentum J of the bound qQ system is given by J=j+S, (9.97)
Heavy Quark Effective Theory (HQET)
310
we note that 2 ( L *S,)
=
1
j(j
+ 1) - 1 ( 1 +
1) -
-1 3
(9.98)
4
2 ( j . s Q > = [ . i ( . i + l ) - j ( j + l ) -3- - ] .
(9.99)
4 Hence for 1 = 0 states, J = S,+SQ, and we have J = 0 or 1. The corresponding 'So and 3S, states ( D ,D*) and ( B ,B') have already been considered. We will suppress the subscript, q [i.e. light flavor index] and concentrate on DJ mesons ( for B J , replace D by B ) for which 1 = 1,j = 3/2 or 1/2. Thus
J =j
+ 1/2, j - 1/2
i.e. we have the states
j = 3/2 j = 1/2
J=2,1
D;, D1
J = 1,0
D;,Do
It is useful to write down the angular momentum part of the wave functions for the four P-states. According to the angular momentum scheme outlined above, P state can be labeled as IJ A 4 j s ~ ) We . can write these states for DJ mesons:
ID* A4
= 0) =
+ 2YlOX+ + K l X +-1 3
1 [Yl-lx:l
0
&
2,
(9.100a) ID1, A4 =
+I)
1
=
- [-Y10xy
fi 1
- [Ylox;'
fi
+ y11 (x: + ZX"]
+ Yl-1 (-x: + 2x!!)]
The P-wave Heavy Mesons: Mass Spectroscopy
1
ID,, A4 = 0)
+ 2YlOXO- + Y,,x;']
Y1-1x:l
=
31 1
(9.100b)
p ; , M = *l)
1
+ y11 (x: + Yl-1 (-x:
= - [-Y10x:l
fi 1 [Ylox;' fi
_ .
ID;,M
d3 [- YI-lx:l
- YIOX-0
= 0) = 1
x"] - x"]
-
+ Y,lx;l] (9.101a)
We first discuss the masses of P-wave mesons. The four P-states are degenerate. The degeneracy between j = 3/2 and j = 1/2 states is removed by the spin-orbit coupling term S,.L in the Hamiltonian Hq given in Eq.(l). Thus we have using Eq. (99): (9.102a) where
5m mj=3/2
=
. +3mDl D2
8
(9.102b) (9.102~) (9.102d)
x
(subscript 1 on refers to 1 = 1 state). The degeneracy between the doublet D$and D1 and the doublet 0; and DOis removed by
Heavy Quark Effective Theory (HQET)
312
the term 0Q.B"in the Hamiltonian (65). For P-wave this term induces the color magnetic moment, interaction of the type
sl2= [12 (S,
+
n) (SQ- n) - 4S, . S Q ]
(9.103)
where n is a unit vector f . Then using the angular wave functions for the states D;, D1, 07, and Do given in Eqs. (101) and (102), we have
2 (s12)D;
=
4 3'
-
(Sl2)0,- 3
(9.104a)
( S ~ Z= ) ~-4. ,
(9.104b)
Hence we can write the masses for these states. We write explicitly the mass formulae for 0; and D1 mesons.
where the parameters til and dl refer to P-state similar to 6 and d for S-state. For m D ; * and moo,, replace $ X I q by - x l g , d~( D ) by 2d, ( D ) and -621 ( D ) respectively in Eq. (107). From Eqs. (106) and (107), we obtain
=
-5 (mn; - mD,) *
(9.108)
Needless to say that for b-flavor P-states, replace D by B and m, by mb. Using the experimental values for the masses, we find
The P-wave Heavy Mesons: Mass Spectroscopy
313
m ~ -; mD, = 40 MeV = m D ; , - m ~ ~Thus , . relation (108) is well satisfied. From Eqs. (108),(70),and (109) we obtain (9.109)
m ~ -moo ; = -200 MeV.
(9.111)
Also from Eq. (106)’ we get
(AIS - A i d )
+f
(xis -
x,,> = mD;, - mD;*
=
113 MeV. (9.112)
On the other hand, for the S-states m D s - m D d = (99.2 f 0.50 MeV) implying that XIs = X l d i.e. independent of light flavor. From Eq. (112), we conclude that if we use the same dl for j = 3/2 and j = 1/2 states then m D ; < mD,. If as expected m D ; > moo, then d1 ( j = 1/2) must have opposite sign to that of 21 ( j = 3/2); in that case there is no reason to believe that they have the same magnitude also. It is hard to imagine that the interaction which removes the degeneracy between j = 3/2 multiplet and j = 1/2 multiplets is strongly dependent on their j values. However, when we take into account the relative motion of heavy quark it is not reasonable to neglect the spin orbit coupling We now take this term into account. Then from the wave functions given in Eqs. (101) and (102) we find
6.
(S-L),;
= 1,
(S
*
1 3
L ) D 1 = --
(9.113)
r)
(S
*
L)D, =
-2.
(9.114)
Thus the contribution of this term can be written respectively for D,*, D1,Q, and Do as: (9.115)
Heavy Quark Effective Theory (HQET)
314
Hence we get
mo; -mD,
=
mDi -moo =
5 2mc 8- dl [I 2m,
+
i:].
(9.116) (9.117)
If we put, z1 = 4dl, i.e. the same strength for the tensor interaction and the spin orbit interaction, we obtain
m q -mo,
32 d i ( D ) 5 2mc
= -___
(9.118) Hence (9.119) Therefore mDi - mD,= 100 MeV
(9.120)
and (9.121) There is no experimental evidence for D;, and Do . Since they are broad resonances, it is not, easy to test Eq. (120) or (121). However if the spin orbit interaction for the relative motion is not taken into account and tensor interaction is not strongly dependent on the j -values, then as we have seen moo-mD; = 200 MeV; hence Do can decay into 0; by emission of the pion and this decay is a P-wave decay and can be distinguished from the S-wave decay Do -+ DT. This is an interesting possibility which can be tested experiment ally.
Decays of P-wave Mesons
9.5
315
Decays of P-wave Mesons
We now discuss the strong decays of P-wave mesons. Parity and angular momentum conservation restricts these decays to the following modes: Dg + (Dn),=,, D; --t (D*n),=,, DI, 0; -+ ( D * X ) ~ = ~ , ~ , and Do t ( D T ) , , ~ .Note that D1,0; 3 D n is forbidden due to parity conservation. It is convenient to express the decay width in terms of the helicity amplitudes (see Eq. 4.41) (9.122) In the rest frame of the decaying particle the helicity amplitudes which contribute are FO,F:, FJ,F;, and F;. In the heavy quark limit the helicity amplitudes are related as follows: j = 3/2 multiplet: n
(9.123) j = 1/2 multiplet: p10 -
-p1f l --po0 .
(9.124)
The simplest way to see this is as follows. The emission of pion by DJ would not affect the velocity of heavy quark. Thus it is the operator S, n which is relevant for these decays. If we select the direction of quantization along z-axis, (i.e. L, is taken along z-axis) then for the helicity amplitudes, the operator S 3 , e Y10 contributes. Then using the wave functions in Eqs. (101) and (102), it is easy to derive Eqs. (124) and (125) by considering the matrix elements of the type
F:
=
f (D*( D )
Y
IS3qY101DJ, A>
(9.125)
where f is the reduced amplitude. Since hadronic decays of Da are pure D-wave, it follows that D1 ---t D*n is also D-wave. For these
Heavy Quark Effective Theory (HQET)
316
decays, therefore F i (s) lps12. Similarly since the decay of DOis pure S-wave, it follows that the decay of 0; is also pure S-wave. The above restrictions are conseqiience of relations (124) and (125) which hold in the heavy quark spin symmetry limit. Hence for the decays Da 4 D x ,D; -+ D*x, and D1 + D*T , we get N
5 IP?rl;mr
M
1.1
(9.127a)
IPTlLr
5 (phase space) 3 -
M
0.44
(9.127b)
where we have used from the experimental data, IpTIDs = 503 MeV, IpnlD*T= 387 MeV, IpslDID.T = 355 MeV. In Eq. (127), the number in parenthesis is experimental value. Thus we see that this prediction of heavy quark spin symmetry is well satisfied. Experimnetally
rD; = r (D; + D*T-+ D*# + DT- + D%O) =
(9.128)
23f5MeV.
From Eq. (128) we get 1 rD 0.31 MeV
(9.129)
rD;
which gives
-
~ D -I
7.1 f 1.5 MeV (18.9+!:: MeV) .
(9.130)
This is in complete disagreement with the experimental value. This shows that the decay D1 + D*x is not pure D-wave; there may be a component, of S-wave. The S-wave widths are usually large,
Decays of P-wave Mesons
317
a small component of S-wave may be possible due to symmetry breaking, since heavy quark spin symmetry is not exact. This may be tested for B,*and B1 decays where the symmetry breaking effects are expected to be small. The decays 0 7 -+ D*x and Do t D n are S-wave decays; thus the decay widths are expected to be large i.e. in the range of few hundreds of MeV. No experimental data are available even on the masses of 0;and DO. To sum up: from the analysis of mass spectrum one cannot conclude that heavy quark spin symmetry is well established; additional experimental data on the masses of heavy hadrons are needed, Except for one prediction on the decay widths viz.
which agrees with the experimental values, the other predictions on the decay widths of heavy hadrons have to wait fot their verification till the experimental data are available.
Heavy Quark Effective Theory (HQET)
318
9.6
Bibliography 1. H. Georgi,”Heavy Quark Effective Field Theory” in Proc. Theoretical Advanced Study Institute (1991) editors R.K. Ellis, C.T. Hill, and J.D. Lykken (World Scientific Singapore, 1992). 2. Riazuddin and Fayyazuddin ”Heavy Quark Spin Symmetry” in Salamfest, eds. A. Ali, J. Ellis and S. Randjbar-Daemi, World Scientific. 3. Mark B Wise ”Heavy Flavor Theory: overview” in AIP Conference Proceedings 302, editors P. Drell and D. Rubin (AIP Press, 1993) 4. M Neubert, Phys. Rep. 245, 259 (1994); Int. J. Mod. Phys. A l l , 4173 (1996). 5 . M Neubert,”B decays and the Heavy Quark Expansion” CERNTH/97-24, hep-ph/9702375, to appear in the second edition of Heavy Flavors, edited by A.J. Buras and M. Linder (World Scientific Singapore) 6. Fayyazuddin and Riazuddin, Phys. Rev. 48, 2224(1993); Mod. Phys. Lett. A, 12, 1791(1991). 7. Particle Data Group, The European Physical Journal C3, 1-4 (1998).
The references to t h e original literature can be found in the above reviews.
Chapter 10 NEUTRINO 10.1 Introduction Experimental puzzles in the past have led to some important discoveries in Physics. Neutrino, which has spin 1/2, was invented in 1930 by Pauli as the explanation of such a puzzle, namely the conservation of angular momentum and energy in P-decay
n+p+e-, require such a particle, so that
n -+ p
+ e- + V e .
(10.1)
Its direct observation was made much later. The electron type anti-neutrinos are thus produced by the decay of pile neutrons in a fission reactor. These can be captured in hydrogen giving the reaction: Ye
+p
+
e+
+
TL,
(10.2)
whose cross-section was measured by Reines and Cowan gezp
= (11 f 2.5)
1 0 - 4 4 ~ ~ ~
(10.3)
to be compared with the theoretical value nth =
(11
1.6)
10-44Cm2.
(10.4)
Note the extreme smallness of the cross-section. It is a reflection of the fact that neutrino has only weak interaction. 319
320
Neutrino
10.2 Mass The question of neutrino mass is one of long standing. In the context of the standard model of unified electro-weak interactions (Chap. 13), there is no understanding of the origin of masses of elementary fermions. In this category the question of neutrino mass also arises. It has an added importance for the following reasons: (a) Among the elementary fermions, only the neutrinos occur asymmetrically in one (LH, left handed) helicity state (see Sec. 11.1.1) i.e. appear to be spinning clockwise as viewed by an observer. This is still an unsolved puzzle. This fact together with lepton number conservation imply that rn, = 0 (see below). However, there is no local gauge symmetry to guarantee the masslessness of neutrino and lepton number conservation in contrast to the photon where both the masslessness of photon and charge conservation are consequences of local gauge invariance of Maxwell’s equations. One may thus expect a finite mass for neutrino. But the intriguing question is why m(v,) << m ( e ) . (b) The interesting phenomena of neutrino oscillations is possible if one or more of neutrinos have non-vanishing mass. (c) Non-vanishing neutrino mass has important implications in Astrophysics. It is a candidate for hot dark matter. It affects history, structure and fate of the universe as we shall see in Chap. 18. Experimentally the question of neutrino mass is still open. This is because (i) m, is small and a smaller quantity is more difficult to measure with high precision than a bigger quantity.
(ii) Neutrino has only weak interaction with matter which implies in practice that no direct measurement of m, is possible.
Mass
32 1
10.2.1 Constraints on neutrino mass a) Direct Limits We first confine to fie * fie comes out in P-decay of Tritium
38
--t
3~~
+ e- + V e .
(10.5)
Electrons from this decay has a very low end-point energy (18.6 keV). As such this process is ideal to look for a possible finite mass of neutrino. If mu = 0, (10.6) Kurie plot is thus a straight line. If mu # 0 ,
This equation also illustrat,es why the end point energy range is important for determining mu = 0. This gives a distortion at the extreme end of the Kurie plot (see Fig. 2.5). Thus one has to look for such a distortion, but note that the deviation is in fact quite small and the experiment is thus quite difficult. An added complication is the presence of final state ionic and/or molecular effects that are not well understood. Anyway, the present limit on v, mass is
mUe< 5.1 eV : m, << me. Direct limits on the other two types of neutrinos are
n p v p : mu,,< 170 keV : mu,,<< m, r + 5nv, : mu7< 18.2 MeV : mu, << mT Thus there is no definitive evidence that v’s have mass, but still the question of neutrino mass is interesting in particle and astrophysical theories as remarked earlier.
322
Neutrino
Figure 1 Basic reactions in double P-decay.
b ) Double P-Decag The double P-decay is another way to look for a finite mass of neutrino. Two kinds of double P-decay can be considered:
( A , 2 ) --+ ( A , 2 + 2) + 2e+ ( A , 2 + 2) + 2 e - .
(2u)
(04
+2 ~ ,
(10.8)
Usually the neutrinos are assumed to be Dirac particles, that is, neutrino u and its anti-neutrino V are distinct. There is another picture of neutrinos, called Majorana in which v and fj are identical. This implies
n
--t
V L + ~ -+
p+e-+Or,=p++-+vL p+e-,
(10.9)
so that (2n) --t ( 2 p ) + 2e- as shown in Fig. 1. The important, physics issues in (OY) double P-decay are:
(i) Lepton number must not be conserved, which is possible if neutrinos are Majorana particles: v = O
Mass
323
(ii) Helicity of the neutrino cannot be exactly -1, this can be satisfied if m, # 0. Thus (0v)PP-decay is especially interesting in determining rn, as half life
T1p o( Q-5 < m,
(10.10)
>-2,
where Q is the Q-value of the reaction involved. There is now distinct evidence of (2v)PP-decay:
"Se
-+
82Kr
T1p
=
(1.1:::;)
x 1020Yr.
(10.11)
Incidently this is the rarest natural decay process ever observed directly in a laboratory. This would help to provide a standard by which to test the double P-decay matrix elements of nuclear theory. From the limit on half-life on (0v)pP decay process 76Ge+76 Se 2e-,
+
T1p> 1.1 x 1025Yr.
(10.12)
(m,) 5 0.68 eV,
(10.13)
This implies which depends on the calculated nuclear matrix elements. Actually, if there is a mixing among neutrinos (see Sec. 4 below), then (m,) = XiIUei12m,,, where X i is a possible sign since Majorana neutrinos are C P eigenstates and U,i arises due to two vertices. c) Astrophysical Constraints As will be shown in Chap. 18, the mass density of all fairly light (m, < 1 MeV) stable neutrinos is
xi
=
C 2 m , ( e ~ )x 2
gm/cm3,
(10.14)
Neutrino
324
where n(1 = 400 cm-3 is the present photon number density. Now the average mass density of the universe is (10.15)
Po = nopco,
where 005 1 and p d is the critical density Pco =
3H; 8nG~
(10.16)
-*
Here Ho is the Hubble parameter, HO = 3 x 10-18ho sec-’, with h, N 0.5 - 0.8 and GN is Newton’s gravitational constant. Thus 2 pco = 2 x 10-29 ho gm/cm3.
The sum of the masses of all stable meutrinos is thus constrained bY 0
P”
I Po
i.e.
C2mvi(ev)x
gm/cm3 5 2 x 10-~~a,& gm/cm3
i
(10.17)
i.e. (10.18) i
The observed age of the universe yields Rohi
5 0.4 so that
Emv,5 40 eV.
(10.19)
i
We may also mention here that, the big-bang nucleosynthesis puts constraints on the mass of any meta stable (m.s) neutrinos which are my,,, (Dirac) > 32 MeV or < 0.95 MeV mv,,,s (Majorana) > 25 MeV or < 0.37 MeV Both the lower limits are in conflict with m, < 18.2 MeV, mentioned earlier, implying that m, must actually be below 1MeV.
Mass
325
10.2.2 Dirac and Majorana masses It is a general feature of weak interactions that only left handed neutrino v~ takes part in it (see Chap. 11). Let us write a Dirac spinor $ as
@ = (5 ) . In a representation in which
75
(10.20)
is diagonal,
$L = 1-2 7 5 . = ( i ) , $ L ? = - - @ 1+75 = 2
( ;) .
The Dirac equation for the two component spinors written as
(10.21)
< and q can be
These equations can also be written in the form iBW&-
mDq = 0
iav,rl-mDj =
o
(10.22b)
where C?=(l,u), 8"=(1,-a).
Under charge conjugation C (particle -iy2@* [see Appendix A], so that
--t
(10.22c)
antiparticle) $ -+ $f =
5 -+I" = -a&? q
For massless neutrino, (22)
t
+ 'Ic =
ia2t*.
(10.23)
and q decouple and we have from Eq.
(10.24a)
Neutrino
326
(10.24b) The plane wave solution of Eq. (24a) is given by
<(4= 4 P ) e-2p.z
- w(p)ei(~'~-Et) -
(10.25)
Then from Eq. (24a), we get [u* p+E] w(p) = 0
(10.26)
with E2 = p2. Let us denote the positive energy spinor by u(p) and negative energy ( E = -\PI) spinor by o(p). Thus we get (10.27) (10.28) where v ( p ) = io2u*(p).Hence we get the important result: if neutrino is massless, we have a left-handed (helicity negative) neutrino and a right-handed antineutrino. This is what is realized in nature. If we start with q-field, then we have opposite case: a right-handed neutrino and left-handed antineutrino. This case is not realized in nature. Let us write = UL and q = U R , then as is clear from Eq. (22) it is the mass which links U L to U R while Eq.(23) can be written as
<
u;
= -ia 2 u1;
u;
= iCT2UL.
(10.29) (10.30)
Hence for a massless neutrino, we will have VL and uk = ia2ui i.e. a left-handed neutrino and a right-handed antineutrino. If we allow both a finite mass and lepton number nonconservation, then for an electrically neutral lepton, the Lagrangian is
L
= \I, (iyPaP- r
n ~Q)
+% (QTC-'Q 2
(10.31)
327
MESS
The second term in Eq. (31) is the Majorana mass term and violates lepton number conservation: AL = 2. Let us define the new fields $1 and $2: 1
41
Jz (E - i & * )
=
-
(10.32) It then follows from Eq. (23) that under charge conjugation $1,2
%,2
(10.33)
i.e. 4 1 , 2 are eigenstates of C with eigenvalues +1 and -1 respectively. In terms of $1 and $2, Eq. (31) becomes [see Eq. (A.107)]
(10.34)
If we start with [ and q or equivalently U L and uR, then we can have two Majorana particles of masses (mD f mM)/2. If we start with V L only, m D = 0, we have a Majorana neutrino of mass mM. In this case Eq. (34) reduces to
L
= zu~cTpd,uL-
3 ( u z ( - i g 2 ) u ~+ h.c.) . 2
(10.35)
We get an important result: a two-component neutrino ( v L ) cannot have a Dirac mass; it can have only Majorana mass, which violates lepton number conservation. Thus one helicity state (- 1 for neutrino) together with lepton number conservation implies that m, = 0. It may be mentioned that if neutrino is massless, there is no distinction between Majorana and Dirac (Weyl) neutrino.
Neutrino
328
Figure 2
Fermion mass generation
10.2.3 Fermion masses in the standard model (SM)and see-saw mechanism The fermion masses in the standard model are generated through a Yukawa coupling of fermions with a Higgs scalar (see Chap.13):
where 4 develops a vacuum expectation value as shown in Fig.2. Here f~ and d) are doublets while fR is a singlet under the standard model group SU(2) @ U ( l ) , e.g.
The above mechanism gives
leading to the Dirac mass mf = 9f
< 4 >o.
(10.38)
Thus rnD(ve)= ge < q5 >o and one thus expects r n ~ ( v e ) rno(t), say within a factor of 10 or so. Also for the neutrino it, is convenient, N
Mass
329
to write the mass Lagrangian in the two-component basis:
= --mD 1 [ Y L N R + Y;N;
+
h.c.1 (10.39) 2 The Majorana mass term can be generated through an effective Lagrangian: a
(10.40) giving a Majorana mass
mu
=
G - (42)o
(10.41)
M
and Majorana
Lm-
+
(v) = - (mV;vL h.c.).
(10.42)
The above mass generation can be pictured as a two-step process shown in Fig.3. This process also gives a Majorana mass to NR
e!:so'ana ( N ) - ( M N i N R + h.c.) . =
(10.43)
We may remark here that with G M 1, .\/z < 4 >o!x 246GeV as in the stanadard model, Eq.(41) gives mu M 10-6eV for M x lof9GeV (Planck mass scale). Referring to Eqs. (39), (42) and (43), the mass matrix in 2 - component basis needs diagonalization. Denoting by prime fields before diagonalization, we have in 2 - component basis
It is useful to consider various limits: Majorana : mD-+ 0 Dirac : m, M -+ 0 Seesaw : m -+ 0, mD
M
Neutrino
330
\.J
I
A
I
‘$L Figure 3
Majorana mass generation
The diagonalization of the mass matrix (44) in the seesaw limit gives
(10.45) Hence we have two Majorana neutrinos UL and N R with masses my mN
N
mL/4M<<mD
N
M.
(10.46)
Depending upon M, U L could be extremely light and N R correspondingly heavy. To summarize, in the Dirac case, one must answer the question why (mvt)Dirac
<< me
(10.47)
while in the Majorana case the seesaw mechanism sidesteps this question; here one has
331
Neutrino Oscillations
by requiring the existence of a large scale M , associated with some new physics. Below we give some typical scales indicative of new physics and the corresponding neutrino masses, which may be relevant for neutrino oscillations (to be discussed below) and dark matter:
10.3 Neutrino Oscillations
If neutrinos are massless, then the neutrinos v,, vp,v,, which enter the weak interaction Lagrangian are also the mass eigenstates. If anyone of them have a mass, then it may be that the mass eigenstates which we denote by vi(i = 1,2,3) are different from flavor eigenstates v,, (w= e, p , 7 ) . In this case, we can get neutrino oscillations. The phenomenon of neutrino oscillations can provide a mechanism to measure extremely small neutrino masses. We note that two sets of states Ivw > and Ivi > are connected with each other by a unitary transformation: Ivw)
(10.49)
= c u w i 14) i
- p i w
u;,
=
(10.50)
sij.
W
Now
H ( k ) I&> = Ei 1%)
Ei
=
(kZ+m:)1/2
(10.51) m2
M
k f L 2k ,
(10.52)
since k >> mi and we take the extreme relativistic limit. Now at time t , (v(t)> satisfies the Schrodinger equation: d
i-Iv(t) >= Hlv(t) > . dt
(10.53)
Neutrino
332
In vi basis, H is a diagonal matrix with eigenvalues E l , Ez and Thus
E3.
Hence from Eq. (49), we can write
and
Thus the probability that a t time t , the neutrino of type w is converted to the neutrino of type w’ is given by
Neglecting CP-violating phases so that to rewrite it as
PW/,= 6,rW - 4
U is real, it, is convenient
UwiUw~jUwjUw~j sin2(.rrL/Xij)
(10.58)
j>i
where L is the distance travelled after which vw is converted to vwl and (10.59)
333
Neutrino Oscillations
where we have used the relation L = ct, X z..7 .
=
2TC Ei - Ej ' (10.60)
As a consequence of CPT and CP invariance
Puw,uw = P-VwI vw - = PUWYWl = p~wDwl.
(10.61)
The form of transition probability (58) depends on the spectrum of Am2 or Aij chosen and the explicit form of U.If Am2 is chosen such that X >> L , then the oscillator term sin2 -+ 0. On the other hand, if X << L , one has a large number of oscillations and sin2 averages out to f , For the conversion of u, to v, (x = p or T ) ,
u=
(
cos8 -sin8
sin8 cos8
and
[
(10.62)
221
pv,+ua= sin228sin2 1.27-L
(10.63)
while the survival probability is Pue+ue= 1 - Pu,+uz. Here 8 is the vacuum mixing angle. Pue-tueand Pve+v, oscillate with L as shown in Fig. 4. The amplitude of the oscillations is determined by the mixing angle; the wavelength of the oscillations is A. To look for oscillations, one needs 0
Low energy neutrinos
0
Long path length
0
Large flux
Neutrino
334
x = ct Figure 4
The neutrino oscillations
10.3.1 Evidence f o r neutrino oscillations One looks for neutrino oscillations in two types of experiments:
(i) Disappearance experiments
+
Reactors are source of Ve through neutron p decay n, --t p + e- Ye and experiment looks for a possible decrease in the Ve flux as a function of distance from the reactor, Ve -, X [if converted to V p , say, one would see nothing, Vp could have produced p+ but does not have sufficient energy to do so].
(ii) Appearance experiments Here one searches for a new neutrino flavor, absent in the initial beam, which can arise from oscillations. All terestrial experiments (except one, see below) are consistent with no neutrino oscillations and provide exclusion regions in the Am2 - sin2 20 plane (see Fig. 5).
Neutrino Oscillations
335
'
10-4
10-3
I
I
;
I
I
lo-2
lo-'
10"
sirhe Figure 5 Plot of LSND Am2 vs sin228 forward region (shaded) limited by solid curves. The region excluded by the BNL E776 ( E N 1 - 10 GeV, L 1 km) and KARMEN up --f ue appearance experiments are bounded by the dashed-dotted and dash-dot-dotted curves respectively. The dashed lines represents the results of Bugey ( E N 5 MeV, L N 40 m) experiments. N
Neutrino
336
There are now three claims of “evidence” for oscillations and hence indirectly for non-vanishing neutrino masses.
a) Los Alamos liquid scintillation dectector (LSND) u-p- - + Ye oscillations In this experiment neutrinos are produced in decays at rest of r+ and p+ : T + --+ p+up,p+ --+ e+vefip,f i p --+ D,. In this experiment E z 30 - 60 MeV, and L 2 30 m. The LSND detector searched for fie by observing e+ as signal in the process Yep --+ e+n. 22 such events were found against the expected background of 4.6 f 0.6 events. If negative results of other expriments are also taken into account, then from Fig. 5, one obtains the following allowed values of oscillation parameters: 7r
0.25 eV2 < Am2 < 2.3 eV2 0.002 < sin226 < 0.04.
(10.64)
These data, which indicate rather large values of Am2 in vp + u, channel, need confirmation from other experiments, e.g., KARMEN which would reach sensitivity of LSND experiment in about 2 years.
b) Atmospheric neutrino anomaly Atmospheric neutrinos are produced in decays of pions (kaon’s) that, are produced in the interaction of cosmic rays with the atmosphere:
p+A
---t
7r*+A’,
7r*
-+
p*up(fip)
--t
e*ve
(fie) f i p (vp>
These neutrinos are detected through the reactions vp+n --+ p-+p, Vu + p + p + + n and v, n --+ e- + p , fie + p 3 e+ + n and are
+
Neutrino Oscillations
337
respectively called p-like and e-like events. One would expect the ratio
However this ratio has been measured in several detectors and it, is found that [MC denotes the Monte Carlo Simulated ratio]
R r ( N ( v p ) / N ( v e ) ) 06s < 1 ( N (vp)/ N ( v e ) ) M c and that it depends on the zenith angle as well, implying neutrino oscillations. The latest atmospheric v data is summarized below:
R
0.63 f 0.03 f 0.05 (Super-Kamiokande sub-GeV) R = 0.65 f 0.05 f 0.08 (Super-Kamiokande multi-GeV) =
The zenith angle distribution of R is shown in Fig. 6. One would not expect up/down asymmetry i.e. between the number of events arising from the neutrinos coming from below the earth and going upward through its center to the detector and those arising from neutrinos coming from up, since we are in a “spherical shell of v’s”. However, for multi GeV one finds for this asymmetry: up/down (e - like) = 0.93 f 0.1 f 0.02 up/down ( p - like) = 0.54-0:0,. +O 06
(10.65)
The former is consistent with no oscillations. The latter is a 6a discrepancy. The result (65) is consistent with v p 1--f v, oscillations which would imply that the former ratio to be unity. The conversion probability P,,-,, as given in Eq. (63) fits the data quit,e well for Am2 = 2.5 x lop3eV2, sin228 = 1.0.
(10.66)
Several long - base line neutrino oscillation experiments that will allow an investigation of the atmospheric neutrino range of Am2 to other channels are at present under preparation.
338
Neutrino
F .!
......., ' ..a.m.
;.:.
'
'
'.A' '1,;' '
( sin'Zd. ~ m ) '
( I
.o. 5x
.
' ' ,012' ' ' 0 : '
10-1
'01.'
)
'
.A'
cxpected zenith angle distribution
-
,A
Super Kimiokondr Prdimino
Sub-GeV
:
Y'J
5, Lf Z6*
1
0.6
-
0.4
i +
0.1
-
O . ,
'
'OS'
I
i.
rj---T-- ....
~
( s'n'2fl. A m ' )
........ -0..
.
I
-0.4
.o.,
0
( 1 .o. 5 0.3
0 4
x lo-' 0.8
) 0"
Figure 6 Zenith-angle distribution of R with neutrino oscillations parameters corresponding t o the best fit values to the Super-Kamiokande data.
( c ) Solar neutrinos
Electron type antineutrinos are produced by the decay of pile neutrons in a fission reactor: R 3 p + e- V , , Electron type neutrinos
+
Neutrino Oscillations
339
are; on the other hand, produced from reactions in the sun, called solar neutrinos. The energy of the sun is generated in the reactions of pp and CNO cycles. Energy is generated through nuclear burning involving the transitions of four protons into 4 H e : 4p + 4 H e
+ 2e+ + 2 ~ +e Q
(10.67)
where Q = 26.7 MeV is the energy release in the above transition. Thus the generation of the energy of the sun is accompanied by the emission of ue's. The total flux of the neutrinos is connect,ed to the luminosity of the sun La by the relation: (10.68) where R is the sun-earth distance, aiis the total flux of neutrinos from the source i, and Ei is the average energy. The most important sources of solar neutrinos in the p p cycle, which dominates cooler stars, particularly the sun, are the following reactions: p p --+ 2He+ve ppe- + 2Hue 7 B e e - ---t 7Liue ' B + 'B,*e+ue
E, < 0.42 MeV : E, = 1.442 MeV : E, 0.86 MeV : E, < 15 MeV :
N
On the other hand, the CNO cycle dominates hot stars and following reactions are sources of ue's:
13N
--+
l50
+ "Ne+ve .
13Cefue
The first reaction in the pp cycle is the main source of solar neutrinos. The third reaction is a source of monochromatic neutrinos. This reaction contributes about 10% to the total flux of solar neutrinos, The fourth reaction contributes only about to the
Neutrino
340
Table 10.1 The standard solar model predictions of neutrino fluxes and observed rates.
Homestake Eth( M e V )
Mode Sensitive to Observed rate
BPSSM (Expected) Ratio
PNUI 0.814 v, +37 c1 + e- +37 Ar ‘ B v ’ s ( 90%) ~
Kamiokande [106crn-ls-] 9.3, 7.5 and 7.0
SAGE and GALLEX [SNU
‘ B v’s
all 3 sources
0.232
but also to 7B, u’s
2.54 f0.20 9.3:;:: 0.273 f 0.03
2.89 f 0.42 +O 25 2.45 f 0.06-0:09 Super-Kamiokande 6.62 f 1.06 0.42 f 0.07 0.368 f O.Ol?::!g Super-Kamiokande
70.3 f7.0 1372;
0.51 f 0.06
interactions per target atom per sec BPSSM: Bahcall-Pinsonneault, Phys. Rev. Lett. 78 (1997) 67. 1SNU
=
total flux but it is the main source of high energy solar neutrinos (up to 15 MeV). Due to different, detection thresholds, solar neutrinos from different sources can be detected in different, reactions. Thus the solar neutrinos with energy > 0.814 MeV can be detected in 37CZ and those > 0.233 MeV in 71Ga. A discrepancy exists between the standard solar model (SSM) predictions of neutrino fluxes and rates observed in terrestial experiments as shown Table 1. We see from this table that in all experiments the observed event rate is
Possible Particle Physics Solutions of Solar Neutrino Problem
341
significantly smaller than the rate predicted by the standard solar model. We may thus conclude that solar neutrinos are detected, thereby establishing the solar fusion. That the observed event rate for solar neutrinos production is smaller than the predicted rate provides a cricumstantial evidence for new physics as will be discussed in the next section.
10.4 Possible Particle Physics Solutions of Solar Neutrino Problem If the experiments are correct, it is very unlikely that non-standard solar models can fit the solar neutrino data. However, there are possible particle-physics solutions, some of which are listed below: (i) Vacuum oscillations (involving 2 or 3
Y’S)
(ii) Matter induced oscillations (involving 2 or 3 v’s) (iii) Sterile neutrino (iv) Magnetic moment transitions Magnetic moment transitions need large neutrino magnetic moment which surpass upper limits on them from astrophysics [see Sec. 51. The possibilities (i) to (iii) involve new physics (nonstandard neutrino properties) in terms of modest extension of the standard electroweak theory in which neutrinos have small masses and lepton flavor is not conserved leading to neutrino oscillations. 10.4.1 Vacuum oscillations Vacuum oscillations of v, to v, give the survival probability : (10.69) where R is the earth-sun distance (cx 1.5 x lOl3crn) while T gives the production location. The results of a fit including all the latest data is shown in Fig. 7. The best fit gives the oscillation parameters
Am2 = 6.0 x 10-11eV2, sin226 = 0.96.
(10.70)
342
Neutrino
(99%) C.L.Allowed Regions
10-10
3 -
n
5PI .
v
2a -
-
!E5lZXl ClAr
+
Kamloksnde
4 experlments
+
+
SAGE
+
GALLEX
Super-Karnlokende
(exp/SSM)
-0.396 f 0.039--
SSM: Bahcall and Pisonneault 1995
Figure 7 Region in the Am2 vs sin228 plane for the vacuum solution in the solar neutrino problem
Possible Particle Physics Solutions of Solar Neutrino Problem
343
10.4.2 Possible explanation in terms of resonant matter oscilla-
tions: Makheyev-Smirnov- Wolfenstein [MS W] efect First we write the Hamiltonian in ve, u, basis [z = p or r or s(steri1e v)] : H,(lc) where H is diagonal in
basis (cf. Sec. 3):
(
sin8
U= M
(10.71)
v1 - v2
and
while Ei
= UHU-’
cos8 -sin0
(10.73)
C O S ~
22k 1 - - Then k + m7 and E2 - El = m2-m2
H,(k)= const.
( y ) +(
-Am2 cos 28 Am’2in28 4k
Am2 sin 28 4k
0
)
(10.74)
where the first part of Eq. (74) is irrelevant for oscillations. Now in traversing matter, neutrinos interact with electrons and nucleons of intervening material and their forward coherent ‘scattering induces an effective potential energy. Such contributions of weak interaction in matter to H , arise due to Feynman diagrams shown in Fig. 8. The first diagram contributes equally to ve, vp and v, and as such is not relevant vp tt vp or v, oscillations. This gives the effective Hamiltonian[see Chap. 131: (10.75) where f = e-, p or n for which respectively I s L = -f, f , -f and Q = -1, 1, 0. The second diagram after Fierz rearrangement gives the effective Hamiltonian: (10.76)
Neutrino
344
I
I
I
I
I
I
; zo
; zo
; w-
I
I
I
Figure 8 Feynman diagrams for neutral current (n.c.) and charged current ( c x ) weak interactions which contribute t o H , for oscillations in matter
where Qe denotes the state of the medium. These diagrams give the potential energy
(10.77)
v,s= 0 where ne denotes the number of electrons per unit volume and n, that of neutrons. Then the Hamiltonian in the matter is [k III E ]
HA4(k)
=
H,(k)+Hw -Am2cQ82e
+
f
i
~
Am2sin2e 4E
Am2 sin 20
0
4E
where
n = ne for ue t--f up or u, =
1
ne - -n, for v, 2
t--f
Y,
1
~
~ (10.78)
Possible Particle Physics Solutions of Solar Neutrino Problem
345
The diagonalization to vl, u2 basis gives:
( ) ( =
CoseM -sinOM
SineM cosOM
) ( L-’ )
(10.79)
with s i n 2 B ~ = sin28-,l M 111
cos 28M = (cos 28 - A ) 1M -
(10.80)
lV
AE
=
E2-E1=-
1 21M
(10.81)
where
E A = 2 h G ~ n m
lv =
E Am2‘
-
(10.82)
(10.84)
For constant density n, the considerations of Sec. 4 give the conversion probability
(10.85) The following are useful limits:
(10.86)
A E = 2fiGFn 7r 8 M = -, u, = v2, u, 2
= U]
Neutrino
346
(iii)
n = nqe, defined by JZG,(n,),,, = 1M
HA4
-+
E Am2sin 28 0
f
(
Am2cos 20 2E
'
Am2 sin 28
Am2sin28 4E
0
(10.87)
Using the above limits, the plot, of E versus n is shown in Fig. 9. Suppose v, is created a t n o > nressay at, the center of the sun, and then it propagates out. If there is no level crossing (shown by dotted lines in Fig.9, then v(n = 0) N v, and undetectable. This conversion of v, into v, is the cause of the depletion of observable neutrinos. Now neutrinos of any energy will not go through the resonance. The resonance condition for any given neutrino energy E is:
n,resE= cos 28
Am2
(10.88)
~&GF '
We may remark here that for ve -+ v p or v, conversion,
(10.89) where Y denotes the number of electrons per nucleon and is 1/2 for ordinary matter. Then, the resonance condition (88) can be written as
=
1
1.3 x 107g/cc- cos28 2Y
[m](7). Am2
MeV
(10.90)
Evolution of Flavor Eigenstates in Matter
347
Ve
= V,
Figure 9 Plot of neutrino energy E versus density n , showing conversion of u, t o ux in matter
For pres 2 p (center of sun) = lOOg/cc, we have
Am2 (&)I1.3 x 105 (eV)2
(10.91)
’
Thus, for example, for Am2 2 6 x 10-6eV2, we will not, have 0.4MeV and the resonance will be at least at resonance for E I E = 0.8 MeV. In this case the resonance will not affect p p neutrino for which Emax= 0.44 MeV but can eliminate 7B neutrinos.
10.5 Evolution of Flavor Eigenstates in Matter The evolution of flavor eigenstates in matter is governed by the equation: (10.92)
Neutrino
348
where H ( z ) is given in Eq.(78). Note that the z dependence arises due to the z dependence of the density n, for varying density case. Using
(10.93) with
U ( X )=
cos e(x) -sinO(x)
sin e(x) COS~(X)
(10.94)
we have
(10.95) where
-
+'' ( 01 2
0
1
) + ( -74)
(10.96)
and 1 0
(10.97)
Noting that, the first part of Eq. (96) is irrelevant, for oscillations and using Eq. (81) we have
For the constant density case, O h (x) = 0 and ZM is independent of x, so that Eq. (98) has simple solutions
(10.99)
349
Evolution of Flavor Eigenstates in Matter
where we have taken z = 0 as the initial point. Then Eq.(93) gives
where we have used the boundary condition v, (0) = 0 [cf. Eq. (79)].Then the electron neutrino survival probability averaged over the detector position L (from the solar surface) is given by
P (ve --+ ve)
+
cos' ev cos' O& sin' ev sin' 1 1 = - - cos2oV cos 2 e ~ 2 2 =
OL
+
(10.101)
where t9v = 0 is the vacuum mixing angle. In general when the density n is a function of x one has to solve Eq. (98) and as a result P(v, --f ve) is given by the Parke formula:
where Bh is the initial mixing angle and Pj Landau-Zener formula. Here
_=
exp (-;y)
is the
-1
and is called the adiabaticity. In the adiabataic limit y >> 1 and Pj--f 0 and we recover the relation (101). The survival probability P ( y e + ve) as a function of E, is displayed for large and small mixing solutions in Fig.10.
350
Neutrino
Large Mixing Solution n
2 t 0.5
s
v
CL 0
1n-'
1
10
lo2
Small Mixing Solution
1.0
f
i
I
1
I I 1111
I I 1
IIll
I
1 1
lllfr
I
f 0.5
s
-.
U
L
I
n
"lo-'
I
I
1 I1111
10"
1
EV (MeV)
Figure 10 Survival probability as a function of small-angle solution
10
lo2
E, for large angle and
Evolution of Flavor Eigenstates in Matter
35 1
( 9 9 % ) C.L. Allowed Regions
ClAr
+ Kamlokande
t GALLEX t SAGE
SSM: Bahcall and PInsonneault 1
stnR(28) Figure 11 99% C.L. allowed regions in the Am2 - sin228 plane for the MSW solution t o the solar neutrino problem
Neutrino
352
We now outline the standard analysis. First determine from Kamiokande the flux of v, from 8 B that reaches the earth. Then one can understand C1 and Ga results with 85
N
78
N
0.40 S S M 0.00 S S M
pp
N
SSM
for the small mixing solution (see Fig.10). One can thus conclude that there exist neutrino oscillations that almost, totally convert, 75, v,’s into u, that, have little effect, on p p v,’s.This leads to the solution shown in Figs.11 and 12, The best-fit, solutions are 1) small-angle MSW:Active Sterile Arn2(eV2) = 5.4 x 3.5 x sin228 = 7.9 x 1 0 - ~ 2) large-angle MSW:
Arn2(eV2) = 1.7 x sin228 = 0.69 3) Vacuum oscillations [cf. Fig. 71:
Am2(eV2) = 6.0 x lo-” sin228 = 0.96 The above different interpretations may be distinguished by new experiments: Super-Kamiokande, SNO, GNO and Borexino. Solar model independent tests of the oscillations may then be feasible. We may mention here that no evidence for “Day-Night” effect
D-N -D-N
-
-0.023 f 0.020 & 0.014
Evolution of Flavor Eigenstates in Matter
353
(99%) C.L. Allowed Regtons I
I
1
I
=
ClAr
+
I
I
I 1 1 1 1
Kamiokande
4 experiments
+
GALLEX
+
1
1
8
1
SAGE
+ Super-Kamiokande 0.400*0.024 (200 days)
SSM: Bahcall and Pinsonneault 1995 10-7
I
I
I I , ,
I
I
I
I I l l
I
1
I
I
i
I I I I
100
sin'( 28) Figure 12 99% C.L. allowed regions in the Am2 - sin228 MSW solution with ye - v, conversion for the solar neutrino problem
354
Neutrino
-3
.
9 - 4
-
N
2
v
w
E
-
0 0
-5 -
-yt -8
~
-4
~
~
~
~
~
~
~
-3.5 -3 -2.5
~
"
-2
~
~
-1.5
~
~
-1'
1
'
"
'
'
-0.5, 0 log(sln 20)
'
"
~
Figure 13 99% C.L. excluded regions found from Day-Night effect for the MSW solution
'
~
"
'
~
Neutrino Magnetic Moment
355
found in Super-Kamiokande has already excluded the heart of the large-angle mixing solution (see Fig.13) In summary it appears that (10.103)
Phenomenological analyses of neutrino oscillations find it difficult to accommodate the above heirarchy of mass ranges in a threegeneration picture unless one of the experiments is sacrificed. For example, if we ignore LSND experiment, a possible solution is sin228, sin228,,
N N
1, Am;, CY 5 x Am:, N 5 x
(10.104)
consistent with the mass pattern m(vT)>> m(v,) 2 m(v,). Some suggest the remedy by introducing a fourth sterile neutrino, which may however, be disfavored by big bang nucleosynthesis (see Chap. 18). We may conclude that neutrinos have masses, neutrinos mix and oscillate, mixing angles are small, solar neutrinos are detected, and the solar fusion is established. None of the above has been convincingly proven. Nevertheless we can say that the neutrino physics provides a circumstantial evidence for physics beyond the standard model. New experiments will test new physics and establish new mass scale(s) indicative of it.
10.6 Neutrino Magnetic Moment With the definition (10.105) where /.LB is Bohr Magneton, magnetic moment interaction is
Hmag= ,U”U B. 9
(10.106)
Neutrino
356
Figure 14 The conversion of VL into V R in the solar magnetic field.
Here B is the solar magnetic field. The neutrino spin would then process in the magnetic field, some left, handed (LH) neutrinos would become RH and sterile to t,he detector as shown in Fig. 14. The conversion probability is determined by
(10.107) Now the solar magnetic field in the convective zone of thickness = (1 - 5) x lo3 gauss, so that the conversion probability is
L x 2 x 108m is B
2 x loam 4 5 . 7 9 x lO-'eV/G)(l - 5) x 103G [3 x 108m/s][6.6x 10-16eV.s] M
~ ( 0 .6 3)1010.
(10.108)
This is O( 1) if K = (0.3- 1)x 10-l' giving p, x (0.3- 1) x 10-lopB. In the standard model,
(10.109) i.e.
357
Neutrino Magnetic Moment
So if p, M 10-lop~, this would definitely indicate physics beyond the standard model. Thus the question of dipole moment of neutrino is very important. What are the other limits on it? The best laboratory limit on m, comes from reactor experiments. In addition to the usual electroweak scattering via W* and Zo bosons exchange, the process Ve
+e
--+
Ve
+e
could proceed via magnetic scattering which is large in the forward direction and for small E,. Consistency with measured crosssection requires (10.111)
More stringent limits have, however, been quoted from astrophysics: (1) Nucleosynthesis in the Early Universe Presence of p, mediates vLe- -+ vRe- scattering. If this occurs frequently in the era before the decoupling of the neutrinos, it doubles the neutrino species and increases the expansion rate of the universe, causing overabundance of helium. To avoid this, pI/ < 8.5 x 1 0 - l ' ~ ~ .
(10.112)
(2) Stellar Cooling Magnetic scattering of neutrinos produced in thermonuclear reactions may occur, flipping the helicity [VL 3 v ~ so ] that the outer regions of the star will no longer be opaque to neutrinos and cooling will proceed much faster. Applied to helium burning star in order that
where Eexotic denotes energy loss due to process of the above types while EH, denotes energy generation rate. This gives (10.113)
Neutrino
358
(3) Limit on pu from Supernova 1987A Neutrinos produced in the initial collapse state have high energies 100 MeV. These high energy neutrinos could escape following spin-flip magnetic scattering [ V L -+ VR]. Furthermore, a proportion can process back V R -+ V L in the galactic magnetic fields and the result on earth could be a signal of high energy (100 MeV) neutrino interactions in the underground detector with a high rate [note that o E2 in V , + p + e+ + n].The observance of no signal implies N
-
(10.114) In view of the above upper limits on p,,, the neutrino spin precession mechanism does not appear to be a viable solution to the solar neutrino problem.
Bibliography
359
10.7 Bibliography 1. T.D. Lee and C. S. Wu, Weak Interactions, Ann. Rev. Nucl. Sci. 15, 381 (1965). 2. R. E. Marshak, Riazuddin and C. P. Ryan, Theory of Weak Interaction in Paticle Physics, Wiley-Interscience, New York (1969). 3. Weak Interaction as Probes of Unification (VPI-1980) AIP Conference Proceedings No. 72 [Editors G. B. Collins, L. N. Chang and J. R. Ficene], AIP, New York (1981), see in particular parts IA and IIA. 4. F. Boehm and P. Vogel, Physics of Heavy Neutrinos, Cambridge Univ. Press, Cambridge, U.K. (1987). 5. R. Eicher, Nucl. Phys. B (Proc. Supp.) 3, 389 (1988). 6. Y. Totsuka, Non Accelerator Particle Physics, In Proc. of XXIV International Conference on High Energy Physics, (Editors: R. Kotthaus and J. H. Kuhn), Springer-Verlag, Heidelberg (1989), p. 282. 7. H. Daniel, Review of Trituim Experiments, in Proc. of XXIV International Conference on High Energy Physics, (Editors: R. Kotthaus and J. H. Kuhn), Springer-Verlag, Heidelberg (1989), p. 1058. 8. J. N. Bahcall and R. K. Ulrich, Rev. Mod. Phys. 60, 217 (1988); see also S. Turck-Chiez et al., Astrophys. J. 335, 415 (1988). 9. R. Davis, Jr., A. K. Mann and L. Wolfenstein, Ann. Rev. Nucl. Parti. Sci. 39, 467 (1989). 10. T. K. Huo and J. Pantalone, Rev. Mod. Phys. 61, 937 (1989). 11. J. N. Bahcall, Neutrino Astrophysics, Cambridge University Press, Cambridge, England, 1989 12. R. Kolb and M. Turner, The Early Universe, Addison and Wesley, California, 1990 13. N. Hata, Lectures delivered at BCSPIN; N. Hata and P. Langacker, Phys. Rev. D56,6107 (1997). 14. J Bahcall, Neutrinos from the Sun, Proc. of the XXV SLAC Summer Institute on Particle Physics, Aug., 4 - 15, 1997, SLACR-528, edited by A. Breaux, J. Chan, L. De Porcel and L. Dixen;
360
Neutrino
Y . Itow, Results from Super Kamiokando, ibid. 15. S. M. Bilanky, Neutrinos, Past, present, future hep-ph/9710251 16. A Balantekin, exact solutions for matter enhanced neutrino oscillation, hep - ph/9712304 17. S. Pakvasa, Neutrinos, hep-ph/9804426, Lectures delivered at the ICTP Summer School on High Energy Physics and Cosmology, June 16 - 20, 1997. 18. Particle Data Group, C. Caso et al, The European Physical Journal C 3 , 1 (1998). 19. M. T. Osaka, Recent results from SuperKamiokande, Repartuer’s talk at 29th International Conference on High Energy Physics, 23-29July 1998, Vancouver, Canada; J. Conrad, Neutrino oscillations, ibid.
Chapter 11 WEAK INTERACTIONS 11.1
V - A Interaction
In analogy with electromagnetic interaction JpAp, Fermi proposed for P-decay the interaction P J F , viz.
+ h.c.
Hint = G [ a i ( ~ ) ~ p Q 2 ( [*~(x)Y’*~(x)] ~ ) ]
(11.1)
The above interaction is for the process 241
+ 3 + ;I; (e.9. n,+ p + e- + ve).
The interaction (1) can be generalized using five Dirac bilinear co-variants. Thus the most general non-derivative foiir-fermion interaction can be written as Hint
=
c
[%(X)riQ2(X)]
[*3(2)ri(cz - C;Y5)%(X)]
a
+h.c.
(11.2)
where ri(i = S, V, T , A , P ) are the five Dirac independent matrices: 1, y p , opV,’yp75, 75. In writing Eq. (2), we have taken into account the parity violation in @-decay. is a solution of Dirac For a massless Dirac particle, if equation, then f y , Q is also its solution. WitJhout,loss of generality, we take only negative sign. Suppose particle 4 is massless, then the bilinear
*
G3(qi% (x)-+ - a3(X)r275~4 (.). 361
Weak Interactions
362
Hence for this case Ci= Ci. Thus we can write Eq. (2) as
If we identify particle 4 with the neutrino, we have the result that only left handed neutrino takes part in weak processes. This is what is observed experimentally (see below). Thus irrespective of the fact whether neutrino is massless or not, Eq. (3) will hold if we take into account the fact, that only left handed neutrinos take part in weak processes. Suppose we impose the chiral transformation for the field a3 viz. @ 3 + -y5@3, then if Hint is to be invariant under such a transformation, we have
cs = cp = CT
= 0.
Hence Eq. (3) becomes
where we put
(11.5) Further we note that if we impose trhe chiral transformation on fields or @ 2 , we have CV
=CA
(11.6)
i.e. E = 1 or V - A theory. We conclude that if one requires invariance of the four-fermion interaction under the chirality transformation of each field separately, we have the V - A theory. We have written Eq. (2) in the order 1 2 3 4. We can go to the order 3 2 1 4 by Fierz reordering theorem: 5
Ki(3214) =
C XijKj(1234). j=l
(11.7)
V - A Interaction
363
The coefficients X i j are given by the matrix
X..23 = --
1 1 4 -2
1 0
1 2
1
-4 (11.8)
where
(11.9) It is obvious that
Ki(3214) = Ki(1432). If we denote by S, V, T , A, P the five quadrilinears appearing in the order (1 2 3 4) and S', V', TI, A', P' when they appear in the order (3 2 1 4), then from Eqs. (7) and (8) we get
V'-A' S'-T'+P'
= =
V-A S-T+P
(11.10)
i.e. these combinations are invariant under Fierz rearrangement. 11.1.1 Helacity of the neutrino To obtain a direct measurement of neutrino helicity, the following reaction was studied .
1 5 2 ~ ~ ( J p = o - ) -+
+
1 5 2 ~ m i , p = , - )V,
1 --+
(152~m(JP=0+) +7) .
The main point of this experiment is that we can select those y rays from the decay of the excited state which go opposite to the v, direction (i.e., in the direction of the recoil nucleus) by having them resonance-scatter from a target of 152Sm.Balancing the spin
Weak Interactions
364
along the upward z direction (v, is assumed to be emitted along this direction), one finds that, the helicity of the downward y-ray will be the same as that for the upward v,. By measuring the circular polarization of y-ray, the experiment, fixed the helicit,y of the y-ray as negat,ive, indicating a lefbhanded v,. Thus it, is established that, only left-handed neiit,rinos take part in weak processes. 11.2
Classification of Weak Processes
(i) Purely leptonic processes The well known example is p-meson decay
In this process four well known particles p-, e - , v,, v,, called leptons, take part. The decay process is described by V- A interaction [cf. Eqs. (4)arid (6)].
c1
(11.1la)
Lr,) and L(,), are lepton currents associated respectively with p meson and its associated neutrino v, and e- and v, LYp) = fi,,YP(1 - 7 5 h L(e)p = Veyp(1 - y5)e.
(11.11b) (11.1lc)
The 7 , and 7 5 ( 2iy0y1y2y3)appearing above are the usual Dirac matrices. We write the lepton current as
L,
=
L’(”,)+ Lye).
(11.12)
Here L f denotes the hermitian conjugate of L,. One can also picture the process (1) as being mediated by a vector boson W,, the so-called weak vector boson. This is shown in Fig. 1 below:
Classification of Weak Processes
365
Figure 1 The muon decay.mediated by a W-boson.
Figure 2
Electromagnetic interaction mediated by a photon
Thus all leptonic weak processes can be described by interaction of the form
where h.c. denotes the hermitian conjugate. Note that Eq. (13) is analogous to electromagnetic interaction of say electron which is mediated by photon and is shown in Fig. 2. The interaction responsible for the process shown in Fig. 2 is the usual electromagnetic interaction
Weak Interactions
366
where a, is the photon field and
is the electromagnetic current:
(11.15)
jFm,= ey,e.
Note the similarity between Eqs. (13), (14), (llb,c) and (15) respectively. Both the electromagnetic and weak currents are vector in character, the appearance of y5 in weak current is due to the fact that parity is not conserved in weak interaction, in fact it, is violated maximally. The coupling of electromagnetic current with the photon is characterized by electric charge (related to the fine structure constant Q by $ = Q = 1/137) while that of weak current with the weak vector boson field W, is characterized by gw (related to the Fermi coupling constant, Gp by =
$
a).
(ii) Semileptonic Processes Some examples of these processes are given below
n + T+
7
-
r
ccco K+ K-
--+
+ -+
+ -+ --f
-+
From these processes, one notes the following rules: 1. The hadronic charge changes by one unit i.e. AQ = fl. 2. In the first four processes, strangeness does not change, in the last four processes it changes by one unit.
367
Classification of Weak Processes
For hadrons, Gell-Mann-Nishijima relation
Y Q=13+-
2
implies that for AQ = f l , either A13 = f l , AY = 0 or A13 = f 1 / 2 , AY = f l , if we assume that AY = 2 processes are suppressed. The processes of first kind are called hypercharge conserving processes and those of second kind are called hypercharge changing processes. In all the processes listed above, we see that either AY = 0 or AY = f l ; no weak process with lAYl > 1 is seen with the same strength as (AYI 2 1 transitions. Thus we have the selection rule AY = 0, f1, AQ = A Y . Since there are so many hadrons in nature, therefore to deal with semi-leptonic decays of each of them would be very tedious. Thus we use the simple picture of hadrons made up of quarks. The main thing about the quarks is that they are regarded as truly elementary similar to leptons. Their weak and electromagnetic interactions would then be like those of leptons. Thus in analogy with Eqs. (15) and (ll),their electromagnetic and weak currents are respectively
(11.17) while
(11.18) where d‘ = cos B,d
+ sin OCs,s’ = - sin t9,d + cos 0,s.
(11.19)
Here 8, is the Cabibbo angle; its value is t9, = 13” or sine, = 0.22. This is introduced since it is seen experimentally that, decay rates for lAYl = 1 semi-leptonic decays are suppressed by a factor of about 1/16 compared to those for AY = 0 processes. We shall
Weak Interactions
368
Figure 3
Quark level process for neutron P-decay.
deal with s’ in Chap. 13. Then in analogy with Eq. (11) or (13) the interaction responsible for fundamental processes like d -+ s --+
u+e-+D,
(11.20)
u+e-+D,
would be
or
Lw
=
-gw JkW-’”
+ h.c.
(11.21)
In this picture neutron ,&decay, A-P-decay and KO 7r++e-+De, for example, would be pictured as shown in Figs. 3-5. Not,e t,he very important, fact, that, bot,h the 1ept)onic and hadronic weak ciirrents in ( l l b , c) and (18) are charged i.e. they carry one unit of charge and t,he hadronic weak currents (18) satisfy the selection rilles IAY I 5 1 and AQ = AY. We also note that in terms of flavor SU(3) notation we can write --f
Jj
= cos BCJ’,
+ sin 8,Ji
(11.22)
369
Classification of Weak Processes
e-
\>
A
Figure 4
Quark level process for neutron A - ,f3-decay.
-I-
Figure 5
Quark level process for
Eo -+ .rrfe-De
Weak Interactions
370
where weak hadronic current is a linear combination of vector and axial vector currents involving respectively y p and ~~y~and are given by
(11.23a) (11.23b) Note also that (11.24a)
(11.24b)
(11 . 2 4 ~ )
Here q =
(
). The heavy quarks and
s' will be considered in
Chap. 13.
(iii) Non-Leptonic Processes Here no leptons are involved. The well known non-leptonic processes are:
( 11.25a)
cc+ -+ c+
-+
-+
nr-(E:) p7r"(Co+)
nr+(CI)
(11.25b)
Classification of Weak Processes
371
(11.25~) or
KO, KO K*
--t
7r+7r-, 7r07ro
-+ T * , T O ,
etc.
(11.26)
Note that all these decays are strangeness changing ([AS]= 1). Let us concentrate on the decays (25), the so-called non-leptonic decays of hyperons. If we consider the decaying particle in its rest frame, the conservation of angular momentum J gives 1 Jj, s - = Jfin;tl= e + s, 2 where C is the relative orbital angular momentum of the pion and the baryon in final state. Since spin s = l/2, l can be 0 or 1. The pion being pseudoscalar (having odd intrinsic parity), the relative parity of final state with respect to the initial state is
Pf
(-l)'(-l) = -1 odd for C = 0 = (-I)'(--I) = $1 even for = I.
=
The s-wave (l = 0) decays are parity violating while p-wave (C = 1) decays are parity conserving. Accordingly decays (25) are governed by two amplitudes, parity violating (s-wave) and parity conserving (p-wave). We can write the Lagrangian responsible for non-leptonic decays as (h)
Lw
- Lh(P.")+ Lh(P.4 - W W - --Jp GF h J hpt h.~.,
Jz
+
(11.27)
where J," is given in (18). The [AS[= 1component of (27) behaves as --sinO,cosOc{i?yp(l GF
fi
- 7 5 ) ~ -) {21yp(l - 75)d).
(11.28)
Weak Interactions
372
Now u and d belong to isospin doublet, I = 1/2 while s is isospin singlet, I = 0. Thus from the combination of angular momentum rules (isospin behaves like angular moment,iim) first, term in curly brackets in ( 2 8 ) has I = 1/2 while the second term in curly brackets has I = 0, 1. Thus the interaction contains both AI = 1/2 and 3/2 parts. Experimentally A 1 = 1/2 part predominates over A1 = 3/2 and then (25a), (25b) and (25c) respectively get, related among themselves. We shall come to these relations later. 11.2.1 p-decay Consider the p-decay p
-
+ e-
+ vp + 0,.
F'rom Eq. (4),we can write the interaction as
(11.29) The interaction writt,en in this order is called t,he charge retention order. It is easier to deal with this order in calculations. Here we have assumed 2-component neutrinos (left- handed vcLand righthanded V e ) but have allowed V -- E Ainteraction, where for V - A , E = 1 and in that, case by Fierz rearrangement we get, Eq. ( l l a ) . From Eq. (29), we can write the T-matrix for p- decay:
T =
&/Z% x [4P2)YA(l - EY5)u(P1)][U(k2)?(1
- %)?@I)] (11.30)
where pl,p2, Icl and k2 are the four momenta of p - , e-, vp and Ve respectively and u(pl),u(pz), u(k l ) and v( kz) are Dirac spinors. From Eq. (30), we get
x (MI2d3p2d3kld3k2
(11.31)
Classification of Weak Processes
373
where
We can easily calculate ]MI2using the standard trace techniques. Neglecting the neutrino masses, we get
Since neutrinos are not observed, we integrate over d3kld3kz. Performing these integrations, and writing d3p2 = 4 ~ p e E e d E ewe , get
where (11.35)
(11.36)
In evaluating the final result (34), we have gone to the rest frame of the muon:
(11.37)
Weak Interactions
374
11.2.2 Remarks (1) It, is always possible to take CV as real and take CA = E C V (e complex).
(2) The electron spectrum does not, distinguish between +1 (V - A ) or E = -1 (V + A ) interaction.
E =
(3) Any deviation from E = f l can be determined by measuring 77 in the electron spectrum. Since 7 is the coefficient of ( m e ( : y E e , > , it plays a minor role except, at low electron energies, where measurements are difficult,. The best experimental value of 7 is
v = -0.007
f 0.013
(11.38)
which is consistent, with zero.
(4) It is instructive to write the electron energy spectrum (34) as
3(W - E,)
+ 2p
-
W
me + 3v-(W Ee
- -2 3m Ee2 )
- E,)
(11.39) where p = 3/4; p is called the Michel parameter. In fact the most general interaction without assuming two-component neutrinos gives the elect,ron spectrum of the form within the square brackets. The experimental value of p = 0.7518 f 0.0026 is in excellent agreement, with p = 3/4 as given by V - E Atheory. We conclude that the two-component neutrino hypothesis is in an excellent, agreement with the experimental results. Finally integrating Eq. (34), we obtain r=7; 1 = G2 ~ P , (11.40a) where ( 1 1.40b)
Classification of Weak Processes
375
If we include O ( a ) radiative corrections
where the fine structure constant a! = -. The Fermi constant G F determined from ( ~ O C ) , using the experimental value for -rp = 2.19703 x sec, is
GF = 1.16637 x
GeVW2.
(11.41)
Decay of polarized muon We have seen that the electron spectrum cannot determine the sign of E . In order to determine E , we consider the decay of polarized muon. Let n, be the polarization vector of muon. We note that
n,2 = npn,p= -1, n, = 0.
(1 1.42)
In the rest frame of the muon m,n,o = 0; thus no = 0 and (0, n). For this case in taking the trace, we put
R
=
Using the standard trace techniques, and performing the integrations over d3k1d3k2, the differential spectrum in the asymmetry angle for p- decay is
(1 1.44) L
where y is the angle between the electron momentum and the p-spin direction and J=-
2 Re E 1 [ E l 2'
+
(11.45)
376
Weak Interactions
It is instructive to write Eq. (44) in the form
(W - E,)
+ 26
-
W-
I)-
1 m2
3 m,,
, (11.46)
where 6 = 3/4 for two-component, neutrinos viz. for V - E A theory. For a general interaction without, assuming two-component, neutrino, the asymmetry distribution in angle y is of the form given within the square brackets. The experimental value of 6 is 0.749 f0.004 in excellent, agreement, with t,wo-component,neutrino hypothesis. The experimental value of E is given by
[PI = 1.003 f 0.008.
(11.47)
11.2.3 Semi-leptonic processes For a semi-leptonic weak process we can write the interaction Hamiltonian as [cf. Eq. (21)].
-Lb’
=
H.
-
-
GF
-JA
Jz
(x)[ E ( x ) yA( 1 - 75) V , (x)]+ h . ~(11.48) .
To first order in weak int,eraction, the T-matrix for a semi-leptonic process of the type
is given by
(11.49) where Ic‘ and Ic are four momenta of electron and antineutrino. We denote four momenta of A and B by p and p’.
Classification of Weak Processes
377
(a) Baryon Decays We consider the case when A and B are spin 1/2 baryons and ( B ) J x [A ) = ( B IV, - AX1 A ) . From Lorentz invariance, the most general structure of these matrix elements is given by [q = p’ - p ]
Since the momentum transfer q = p’ - p is very small compared to the mass of A or B for the processes we are considering, we can write
Now we shall take A and B as members of the spin 1/2 baryon octet and then
J~ =
COSO,
JX+ +sino,
J:,
J:
= COSO,
J,- +sinO, J:+, (11.52)
J i , Jit are 1 f 22 and 4 f25 components of octet of
where J f and currents J ~ (2x = 1,-
- . ,8). As shown in Chap.
5
Weak Interactions
378
where g x k = if i j k F
Since F,
=
+ dijkD.
(11.53b)
J KO((),x) d 3 2 is a generator of SU(3), it, follows that
where
gvijk = ifijlc.
(11.54b)
Thus if we neglect the momentum transfer q 2 , (q2 M 0), the matrix elements (BI,IJix( B j ) are essentially determined in terms of Cabibbo angle 0, and the two reduced matrix elements F and D. Using Eqs. (53) and (54), the matrix elements of these decays are given below: Expt,. Decay Vector Axial vector Ratio value of B+B’lv, current gv current g A gA/gV
-0.340 f0.017 4312 sin 0, F-;D 0.25 Z- 4 A fisino, f0.05 x ( F - ;D) In order to test, the octet; hypothesis, we note that if we determine F and D from the first t,wo decays, we find F - D and F - ; D for the third and fourth decays in agreement, with their experimentdl values. The parametrization given in Eqs. (53b) and (54) is in excellent, agreement with experiment. Using the first two entries of the above table, we find F = 0.444 f 0.015, D = 0.823 f 0.015.
C-
-+n,
- sin0,
sin0, ( F - 0) F - D
Classification of Weak Processes
379
As an example to show how gA/gV is determined we consider the case of neutron p- decay n + p e- Ve in detail, where from Eqs. (51) and (52) we have
+ +
[gVyfi- gAyfi’Y51u(P)
(11.55)
+
with gv = cos 8,, g A = cos 8, ( F 0 ) .In the rest frame of neutron, we write k‘ G (Ee, p,), k (E,, p,), p = 0, p‘+ pe+ p, = 0. Since q is very small as compared to neutron and proton masses, we can treat them non-relativistically. Then
Let us write the leptonic part as
LP = f i ( k / ) - y f i ( l - - y 5 ) ? l ( k ) .
(11.57)
The amplitude F [cf. Eq. (49) and Eq. (2.75)] is given by
(11.58) We now sum over proton spin and lepton spin and define the neutron spin Sn as
1 S n = ~ x n +a X n .
(11.59)
Using Eq. (2.110), we get for the probability distribution
(11.60)
Weak Interactions
380
where
8,
is the direction of the neutron spin and
A = A =
A’
=
B =
D = The experimental data give the following values of these correlation functions,
X = -0.102 f 0.005 A’ = -0.1162f0.0012 B = 0.990 iz 0.008 D = (-0.5 f 1.4) x
If we write
J: =
( 11.62)
( g A / g V ( ,t,hen the value of X gives Z=
1gA/gvl = 1.261 f 0.004.
(11.63)
The very fact that, B is nearly 1 implies the maximum parity violation in ,&decay. The value of A’ [assuming gv and g~ are relatively real, see below] gives IgA/gvI = 1.267 f 0.014, consistent with Eq. (63). A non-zero value of D would imply t h e reversal violation in P-decay. The experimental value of D is nearly zero and show that time reversal invariance holds. If we write g A / g V = -zei6, where for q5 = 0 or T , T invariance holds, we obtain q5 = (180.07 f 0.18)’.
(11.64)
Finally, from Eq. (60), we obtain for the decay width I‘: (11.65)
Classification of Weak Processes
381
where
and f(po) =
1''
d p p2
(JGJG) = p y . -
PO
-
(11.66)
me
Since charged particles are involved, this expression of f r = fr-' is subject, to radiative corrections, which are normally incorporated into the factor f along with the first, order Coulomb corrections. These corrections change f by about 5%. The average value from direct neutron lifetime measurements is r = 888.6 f 3.5 sec.
(11.67)
Knowing [gA/gV1 , one can determine Gv = GFgV from Eq. (65). Gv can also be determined from the superallowed O+ Of pure Fermi decays for which FT value is --f
(11.68a) where (11.68b)
F here is different from f for the neutron p- decay and it must account for the stronger Coulomb effect, and for the much more subtle radiative effect,s associated with the higher electric charge. The quantity ( F T )=~3070.6 ~ f 1.6 sec from the O+ + O+decays together with the phase space factor F from Wilkinson and the value of ( g ~ / g v given I in Eq. (63) gives T = 894 f 37 sec, to be compared with the direct neutron life-time measurement given in Eq. (67).
Weak Interactions
382
Finally the Cabibbo angle (11.69) where GL = 1.16637 (13) x GeV-2 while AD and A p are the “inner” radiative corrections to both nucleon and muon AD-decay with Ap - A p = 0.023 (2). This gives
IVud(= COS~’, = 0.9744 & 0.0010.
(11.70)
sin Bc is determined from hyperon P-decays and is given by
lVusl = sine, whereas from
Ke3 decay
=
0.2176 f 0.0026
(11.71)
its value is 0.2196 f 0.0023
(b) Pseudoscalar Meson Decays
(i) Pion Decay 7r-
-+ c-
+ v,,
.!?= e,p.
For this decay, the T-matrix is given by
T
GF
= --
Jz
Here, we have p = k’
COSB‘
(0
p,+I
7r-)
+ k . Now from Lorentz invariance
Using the standard techniques of Chap. 2, the decay rate r can be easily calculated. We obtain
r (*- + e- + ve) = G$ cos2 87r Oc
f: m: m, (1
-
$)’.
(11.74)
Classification of Weak Processes
383
It thus follows that pion decays mainly to muon, its decay to electron is suppressed by a factor rn:/rn; (phase space). In the same way, we can write down the decay rate of K !+ Q; it is given --f
by
r ( K - -+ e-
2) 2
8, + ot) = G$ sin2 8n
fimi m K (1 -
. (11.75)
born the experimental values of the decay rates for pion and kaon we can determine fir and f ~We , get fn M 131 MeV and f K / f i r x 1.22. Fkom the particle data group: f, x (130.7 f 0.1 f 0.36) MeV, fK M (159.8 f 1.4 & 0.44) MeV. Remarks Suppose pion decay occurs through a vector boson W . Then we can write the decay amplitude F :
F = -gw i f n f l
-gpx
+w
yw
(11.76)
ii(k’) yx (1 - y5) ‘u (k). P2 - mw We write the W-propagator in the following form 1 p2
- m&
[(-%A
+
y )+ 1
p 2 - rnb PpPA
P 2 mi/
PpPx
I (11.77)
The first part of Eq. (77) gives the transverse part of the propagator and second part gives its longitudinal part. If we substitute Eq. (77) into Eq. (76), we find that the first part of Eq. (77) gives zero and the entire contribution comes from the second part. We get
2
=
--gw if, me m2w
qc’)
(1 - 75)21 (k).
(11.78)
Weak Interactions
384
Here we have used the Dirac equation U ( k ’ ) ( y . k’ - me) = 0 and p = k’ k. Thus we note that the longitudinal part behaves as if the decay has taken place through a scalar particle of zero mass with effective coupling g&/mk. We also note that it gives a contribution proportional to the lepton mass which is reflected in the formula (74). This is called helicity suppression.
+
(ii) Strangeness Changing Semi-Leptonic Decays As an example of these decays we consider the decay. K-
+ no
+ t- + P!,
We first, note the rule: Ad)
T
=
GF ---sinQ,
fi
= AS =
( = e,p.
1 . The T-matrix is given by
( noI Jl i -K )
The Lorentz striictme of the hadronic matrix elements is given by
(no
IpiI K - )
=
(no
- -
1v;I K - ) 1
1
(W3J2pn2pb x [f+ ( q 2 ) ( P + P‘>x + f- (a2) ( P - ?%] ?
( 11.80) where p and p’ are four-momenta of K - and T ” , q = (p’- p) and k’ and k are four momenta of e- and ve respectively. In the rest frame of K - , we have m K = w Ee E,, pT+pe p, = 0. Using the standard techniques of Chap. 2, we get
+ +
dr 1 = -G$sin28, dEP dw 4n3
~
If+
+
( q 2 ) I 2 [A+BRe[+CI[12],
(11.81)
385
Classification of Weak Processes
where
C =
w = t
=
-41[ w - w ]m;
mK+m,-m; 2 2 2 mK
f- (Q2) / f+ ( q 2 )
(11.82)
For electron, we can neglect, its mass i.e. we put, me2 M 0. Then Eq. (81) is much simplified. In this case, we get, for the electron spectrum
Here we have put
f+ (q2) M f+ (0) = f+. For this case we obtain (11.84)
In the SU(3) limit (7ro IV'l K - ) 0: if4+i553 so that f+(O) = Consider the neutral Kaon decays:
KO KO
-+
7r-+e+ fve,
-+
7r++e-
tve,
5.
AS=AQ AS=-AQ.
For the first case the hadronic matrix elements are given by
Jlt creates negative charge and S = -1.
For AS = -AQ, no such current can be written down in this conventional theory. For more details for semi-leptonic K-decays see Ref. 2.
Weak Interactions
386
Hadronic weak decays (a) Non-Leptonic Decays of Hyperons l l .Z.4
Consider the decay
B (P)
+
B’ (P’) + 7r ( k ) ‘
The Lorentz structure of the T-matrix for this process is given by
The amplitudes A and B are functions of scalars: s = (p’+ l ~ ) t ~=, ( p - P’)~. A is called the parity violating (p.v) [or s-wave] amplitude and B is called the parity conserving (p.c) [or p-wave] amplitude. In the rest frame of baryon B p’ = -k,
lp’l = Ikl = k ,
p’ = kn,
(11.86) In this frame, the amplitudes A and B are constants. In the rest frame of B
(11.87)
where x is a constant 2-component spinor. Using Eq. (87), we may write the T-matrix (11.88a) T = x’ M
x,
Classification of Weak Processes
387
where (11.88b)
We note that the p.w. amplitude A is essentially the s-wave amplitude and the p.c. amplitude B accounts for the p-wave amplitude. The decay width is given by
d r = (24’
s4( p - p’ - IC) [-TT-( M M t ) ]d3p‘
d3k.
(11.89)
Performing the integration, we get the decay width
li Pb [ / a s /+ r =2I~~I’] 21rm a
(11.90)
We now consider the decay of polarized baryon €3. Let S be the POlarization (spin) of B. Let s be the polarization of decayed baryon B’. In the rest frame of B‘, s gives the spin of B‘. The decay probability in this case is given by
dw
=
pn17s4 ( p - p’
- IC) x -1 { T r [ ( l + u ~ ~ ) M ( l + a ~ S ) ] M ~ } d ~ p ’ d ~ k .
2 (11.91) The trace can be easily evaluated and the transition rate is proportional to
where a=
2Re a,* up b S l 2
+ l%I2 ’
P=
21m a,* up b S l 2
+ bPI2
Weak Interactions
388
r=
laJ2 - b P I 2 lasI2
Q2
+ bPI2
+ p2 + y2 = 1.
(11.93)
Because of the last constraint, we can write
p
=
(1 -
sin4 1I2
y = (1 -2) cos$
4
=
tan-'
(P/r).
(11.94)
One also defines
A
= -tan-'
(D/Q).
If we do not, observe the polarization of B', we put, s
=
0 and we
get
dW/r
=
dQ.9
-[I
47r
+
Q
S n] .
(11.95)
Hence we can write the angular distribution
I B (8) = Const [ 1 + a S cos 81 ,
(11.96)
where 8 is the angle between the hyperon spin S and the decayed baryon momentum direction n. If a = 0, the angular distribution is isotropic. Q = 0 implies either a, = 0 or ap = 0. For this case parity is conserved. The anisotropy in angular distribution implies nonconservation of parity. From the angular distribution we can determine the product, as. Since the polarization S of baryon is not generally known, it, is difficult to measure Q! by this method. Further information about Q can be obtained from the polarization of decayed baryon B'. From Eq. (92), we obtain the polarization of decayed bayron B'.
(11.97)
389
Classification of Weak Processes
In particular if the original baryon B is unpolarized viz. S = 0, we get (s) = a n. ( 11.98) This equation implies that the baryon B’ obtained from the decay of unpolarized baryon B is longitudinally polarized. Thus a measurement, of this polarization allowed a direct determination of a. The experimental values for a, ,!l and y are given in the Table 1. Now a non-zero value for ,8 implies the violation of time reversal invariance in these decays. From Table 1, it is clear that ,Ll = (1- (Y’)~’~ sin $ is consistent with zero. Thus the time reversal invariance holds in these decays. P invariance implies either a, = 0 or up = 0, so that Q! = 0, p = 0. But Table 1 shows that a is non-zero. C invariance implies a = 0, ,f3 # 0; hence from Table 1, it follows that C invariance is also violated. The consequences of T and C invariance quoted above hold if we neglect the final state interactions.
(b) A1 = 1/2 Rule for Hyperon Decays The effective weak Hamiltonian responsible for ( A S ( = 1 nonleptonic decays in the conventional theory is given in Eq. (28), namely
G F sin Bc cos Bc - -
--fi
H,,
where
Hw
+ +;.
=
[JT (J”)’
+ h.c
(11.99)
*I J i .
(11.100)
+
NOW JX+ U 7~( I 75) d has I = 1, 1 3 = +1, S YA (1 75) u has I = f, 1 3 = Thus in general Hw has a mixture of A 1 = 1/2 and A 1 = 3/2. However, the most striking effect, of these decays is the approximate validity of A1 = 1/2 rule. The decays N
N
Weak Interactions
390
Table 11.1
Y
Decay
'! '! A pr-
4
"' *0
Q
dJ
0.642 & 0.013 (-6.5 f 3.5)'
A
(derived) (derived)
0.76
(864)'
0.65 f 0.05
-
-
-
+0.017 -0.980-0,0,5
(36 f 34)'
0.16
(187 f 6)'
':nr+ ' +'
0.068 z t 0.013
(167 f 20)'
0.97
(-73';F)O
:
-0.068 f 0.008
(10 f 15)'
0.98
(249':;o)'
-0.411 f 0.022
(21 zt 12)'
0.85
(218?ii)o
f 0.014
(4 f 4)O
0.89
(188 f8)"
--f
nr
c; : c+ --t
pro
'-
nr= a . 0' .zo -+
Y
- -_ -0.456
-+A+r' .. -+A+;?-c _ Y_
v
39 1
Classification of Weak Processes
with A1 = 3/2 are suppressed. A satisfactory understanding of this rule is still lacking. We now examine the consequences of A 1 = 1/2 rule in nonleptonic hyperon decays and its approximate experimental validity. Consider first the decays
A! A:
A13=1/2 A13 = -1/2
: A-+p+n:
A+n+no
AI
=
l / 2 , 3/2, *
* *
The simplest, possibility is A I = 1/2. Assuming this to be the case, the only possible isospinor which one can form is
fi T * T A = ( p
no + h fi r-, & p 'n
- fi
no)I\. (11.101)
Then for AQ = 0, we have
AO_ = -&A:.
(11.102)
Hence we get
r (A:)
= 2 r (A;) = "At.
"A!
It is clear from Table 1,
x
Q
~
O
,
(11.103a) ( 11.103b)
; experimentally
(11.103~) Thus AI = 1/2, rule is a good approximation, AI = 3/2 amplitude is very much suppressed for A-decays. An exactly similar argument gives z- = z;, (11.104a)
--
-Jz
which implies
r (z:) /r (z:)
=
a=-/a=o =
-_
-0
2 Expt : 1.639
( 1 Ezpt : ( 0*456 0.411
M
1.11). (11.104~)
Weak Interactions
392
For C-decays, assuming A1 = 1/2, the only isospinors which we can form are U
N
(z'.7r)+ibN
(11.105a)
(CX7+7.
Writing only the part, for which total charge is zero, we have
(C-T' + C'T- + COT') + b (fip COT' - n c+T- + n C + K ) .
u fi
-
h j3 C'7-r' (11.105b)
Thus we get
CI = ~ + b
CI
= a-b
Co
=
f i b
Ci
=
-JZb.
(11.106)
From Eq. (106), we get
CZ
-
C I = JzC,+.
(11.107)
The prediction can be tested as follows: In the ( u s , u p )plane if we regard C:, C: and d C , ' as vectors, then they should form a closed triangle. To sum up, in case the AI = 112 rule holds, out of 7 decays listed in Eq. (25) only four are independent. In the language of flavor SU(3) [cf. Chap. 51, the dominance of A1 = 1/2 rule is generalized to octet dominance. This can be seen as follows: u,d , s, belong to 3 representation of SU(3). ti, d, 3, belonging tJo3 representation of SU(3). Now 3 @ 3 = 8 @ 1. Thus J," in Eq. (27) belongs to an octet representation of SU(3). Hence Hkt in Eq. (27) or (28) contains 8@8=1@8@8@10@+TO@27.
393
Classification of Weak Processes
Figure 6 W-boson exchange graph for the reaction u + s -+ d + u
.
It can be seen that only 8 and 27 are relevant for the decays (25). Thus HLt contains both 8 and 27 where 8 corresponds to AI = 1/2 only while 27 contains A1 = 3/2 as well. Thus in the language of SU(3), generalization of A1 = 1/2 rule is the octet dominance. The oc'tet dominance for the current-current interaction implies an additional relation (called Lee-Sugawara relation) between s-wave decay amplitudes of (25)
+
2A (Z) A (A!)
(c) Non-leptonic
= +fi A (C:)
.
(11.108)
Hyperon Decays in Non-Relativistic
Quark Model One can recover not only the AI = 1/2 rule but also the right order of magnitude of the scale required to reproduce the s- and p-wave fits of non-leptonic hyperon decays. Consider the weak vector boson exchange graph of Fig. 6 as the analogue of the gluon exchange quark-quark scattering graph considered in Chap. 7 which quite successfully described the quark spectroscopy. The matrix elements for the process shown in Fig. 6 are of the form
Weak Interactions
394
where q = pi-pi = pi - p j . u's are Dirac spinors in Dirac space but are column vectors involving u,d , s quarks in ordinary flavor SU(3) space. a; and ,Bj'are operators which transform a v-like state into a d-like state and a s-like state into a u-like state respectively. We take the leading non-relativistic limit, of the above matrix elements. In the leading non-relativistic approximation, only yo and yi 7 5 have nonzero limits. Thus only parit,y conserving ( p . c ) part of A4 survives in the leading non-relativistic approximation and we have in this limit
(11.110)
MP'"
= 0.
The latter corresponds to a general result that (B' I(JJ>"'"I B ) = 0 as a consequence of CP and SU(3) invariance. The Fourier transform of Eq. (110) gives the effective Hw as
x (1 - ai.aj)f i 3 (r) .
(11.111)
Now it has been shown [see Sec. 12.4.21 that, in the currentalgebra approach the question of AI = 1/2 rule or octet, dominance for non-leptonic decays of baryons hinges on the matrix elements (BS
lHFl
BT)
aTSUu,
( 11.112)
which essentially determine both s- and p-wave amplitudes. Here u is a Dirac spinor for B, or B, which denotes a baryon like A, C, E,n, or p . Therefore, we have to take the matrix elements of Eq. (111) between the baryon states B, and B,. We regard the baryon state B, or B, as made up of three quarks. We take the
Classification of Weak Processes
395
spatial wave function for such states to be the same for the octet of baryons p , 71, A, C*, C‘, So, E- and denote it by Qo. Thus writing df = 1 6 3( .)Iq 0 )= p0( o ) I ,~ (11.113)
po
where r = ri - rj (i # j ) , we have to calculate the matrix elements of the operator
i>j
’
between the spin-unitary spin wave functions of the states p , n, C+, Co,A,Eo,given in Chap. 6 . We obtain (11.114a)
agoho
a2-z-
=
GF sin 8, cos &d’ (- 6) t/Z
(11.114b)
=
-d5apn
(11.114~)
=
GFsin8,cos8,df (-2d6)
(11.114d)
Jz
= 0.
(11.114e)
= -&upn expressed in Eq. (114c) enThe relation sures the A1 = 1/2 rule (or octet dominance) and hence A (CT) = 0 (which is good experimentally) in current algebra approach [see Eq. (117) below]. Once the octet dominance for ars is established we can parametrize ars in the SU(3) limit, as
Then the relations (114) immediately give
D’- -1. F‘
( 11.116)
Weak Interactions
396
Now using the current algebra relations [see Sec. 12.4.21 for the s-wave amplitudes one has
A(C$)
A
(c:)
=
1 -
=
-f f f ( U P p
d5 fir 1
+ d5 axon) (11.117)
+
Here f T is the constant which enters in T - t pfip decay. Then using Eqs. (114) and (117), we have the relations (107) and (108). Using the value of d’ as determined by the constituent quark spectroscopy [cf. Chap. 71 ,
ax+p
-27 G F sin BC cos Bc =
(mE - u L A )
8&.rras x -105 eV
(
7h2 A
1 - m/ms
)
constituent
(11.118)
for the accepted value of as (q2 M 1GeV) M 0.5.This is almost the phenomenological octet dominance scale, which together with D’/F’ M -0.86 [not, very far from the prediction (116)], are required to fit, the s- and p-wave amplitiides of hyperon decays.
11.3 Problems (1) Show that the electron spectrum in the decay of b-quark b using V
-
--f
c
+ e- + v,,
A theory is given by (neglecting the electron mass)
Problems
397
where y = - 2Ee ,
? J m = l -m,2 -. mb mi Similarly, show that for c-quark decay c
+s
+ e+ + v,
the electron spectrum is given by
dr _ -
dx
Hint: For b -+ c + e-
-G2F 5 1 6 7 ~mc ~
2(Ym
- YIZ
+ Pe, the matrix elements are
Use Eqs. (31) and (32) with the replacements (mp,me,mum,mve) (me,m,, me,mUe), E = 1 so that --f
in the rest frame of b. Performing d3p2 integration, write d3kld3k2 = kf dkl dlc2 d R and use
to perform the angular integration to obtain
398
Weak Interactions
where from
E, =
2 mb - mz - 2 mb Ee 2 (mb - me + Eecos8)
one has
s.
The int<egrationof Eu gives the result, For the second problem, the matrix elements are
GF
-
T=----[ 4 P 2 ) Yx (1 - 75) '11(P1)1 [u(ka) Yx (1 - 75) 4w] '
Jz
Results from the first can be obtained by changing k:! t-$ m,-+ms
h,mbtmc,
and then follow the same steps as in the first part. ( 2 ) Consider the decay
K
-+
37r.
Show that decay rate can be expressed as
o<x2+y2<1, where A is the decay amplitude,
Problems
399
Tl,T2 and T3 are kinetic energies of pions. Then the energies w l , w2, w3 of pions are given by wi = Ti+m, and Q = TIt T 2 +T3 = w1 -k w2 + w3 - 3m, = mK - 3m,. The events in Dalitz plot can be expressed by taking
where j stands for any decay channel of K. (3) Show that if the three pions in the decay of K in I = 1 states, then
r (K;
=
.+
-+
3n are
r+r-ro)= 2r (K+ .+ r+roro)
r ( K + -+ r+r+r-)- r ( K + r ( K ; -+ ronono).
t
(1)
r+roro) (2)
Equations (1) and (2) are the necessary conditions for AI = 1/2 rule to hold. But they are not sufficient since I = 1 state can be reached also by A1 = 3/2. Show that for totally symmetric I = 1 states
r ( K + r f r + ~ - =) 4r ( K + + r+ro?To), 3 r ( K ; -+rororo)= -r ( K ; ,+r-ro). 2 .+
-+
Weak Interactions
400
11.4
Bibliography
1. T. D. Lee and C. S. Wu, Weak Interactions. Ann. Rev. Nucl. Sci. 15, 381 (1965).
2. R.E. Marshak, Riazuddin and C. P. Ryan, Theory of Weak Interactions in Particle Physics. Wiley-Interscience, New York (1969). 3. L. B. Okun, Leptons and Quarks, North-Holland Publishing Co., Amsterdam, (1982). 4. E. Commins and P. H. Bucksbaum, Weak Interaction of Leptons and Quarks, Cambridge University Press, Cambridge, England (1983). 5. H. Georgi, Weak Interaction and Modern Particle Theory, Benjamin/Cummings, New York (1984). 6. T. D. Lee, Particle Physics and Introduction to Field Theory, Harwood Academic (revised edition 1988). 7. Particle Data Group, The European Physical Journal C3 1-4 (1998). 8. S. Reedman, Comments on Nuclear and Particle Physics, Part A, Vol. XIX ( 5 ) , p. 209 (1990).
Chapter 12
PROPERTIES OF WEAK HADRONIC CURRENTS AND CHIRAL SYMMETRY 12.1 Introduction In Chap. 11, we have introduced an octet, of vector and axial vector currents (12.1) (12.2) where
Jxf
=
Ji,Ji+=
KGZX + Alizizj, b i 5 X
+ A4fi5X
(12.3) (12.4)
take part, in lAYl = 0 and lAYl = 1 semi-leptonic processes respectively. The electromagnetic current is given by
VTrn=
v,,+
1 -&j,
fi
(12.5)
where the first part is the third component,of an isovector while the second part, is an isoscalar. Now HfZ N Viema,.Since photon field ux has C-parity - 1 and the intrinsic parity of the photon is - 1, we see that CP of Vfm is +l. F'rom this we can generalize that CP of vector current Vj, is +l. The parity of axial-vector current Ax is +l and since the weak Hamiltonian is CP invariant, the C-parity of AA must be f l . 401
402
Properties of Weak Hadronic Currents and Chiral Symmetry
12.2 Conserved Vector Current Hypothesis (CVC) The hypothesis of conserved vector current (CVC) states that V : and fix(= Jim, A 1 = 1) are respectively 1 22, 1 - 22 and 3 members of an isospin current, which is conserved by strong interaction. The generators of the isospin group SUI(2) are then given by
+
I2 =
s
Ko(x, t ) d 3 x , 2
=
1, 2, 3.
(12.6)
The first consequence of CVC (a’Vv,’ = 0) is that the form factor hv(q2)= 0 in Eq. (11.50a) where A and B are respectively taken as neutron and proton. [Note: When invariance under SU(2) is assumed, mp= m, = mN.1 In order to discuss the other consequences of CVC, we note from Eqs. (6) and (11.50a) t,hat,
(12.7) Since I+ is conserved in the absence of electromagnetism, I+(t) is a constant of motion i.e. I+(t) = I+(O) = I+, we can take t = 0 and
(12.8) Now
and thus
Conserved Vector Current Hypothesis (CVC)
403
Hence it follows from Eq. (7) that
gv(0) = 1.
(12.11)
Thus in the absence of electromagnetism, the vector coupling constant in nuclear ,@decay is not renormalized and is equal to its "bare" value. Noting that [ J r = ~ V S inX SU(3)]
(12.12) and
I+ I4 = lP) (PI I+ = (4 I
(12.13)
it follows that
(PIv,ln )
= (P I[V3x, I+]]n)
(P I [Vxem, 1+1 I4 = ( p lVfrnlp ) - (TI, lVfemlTI,) . =
(12.14)
Now Lorentz invariance gives the electromagnetic form factors of proton and neutron as (P(P'> lVxenlP(PU
(12.15)
[ 12.16)
404
Properties of Weak Hadronic Currents and Chiral Symmetry
where [since J d32 ~ “ ( x0), is the electric charge in unit of e ] it, follows, on using Eqs. (8)-(10) that
F,P(O)= 1, F;”(O)= 0.
(12.17)
Since gxVqv gives Pauli type interaction, it also follows that
F g o ) = K p , F,”(O) = K ,
(12.18)
where K~ and K , are the anomalous magnetic moments of proton and neutron respectively. np = 1.792 and K , = -1.913 in units of nuclear magneton. Hence we get from Eq. (11.50a) and Eqs. (14)-(16) that
where F y and F . are the isovector electromagnetic nucleon form factors. Their normalization follows from Eqs. (17) and (18).
Thus in particular
(12.21) Using SU(3), we can write the matrix elements of vector current &, i = 1, . * * , 8 for an octet of baryons (assuming q2 x 0):
namely the relation (11.54)
Partially Conserved Axial Vector Current Hypothesis (PCAC)
405
12.3 Partially Conserved Axial Vector Current Hypothesis (PCAC) Fkom Eq. (11.73), we have
(0 l t P ~ z ( x )r-) I
=
-ipA (0
IA;~
r-) e-ip'x
- - 1 -f,m,e 1 (2743/2 &
2 -ip.x
. (12.23)
If the axial vector current A: is conserved, then either f, = 0 or rn; = 0. Since for a physical pion rn; # 0, f r r must be zero and pion decay is forbidden. Thus A: is not conserved. Now @A: has the same quantum numbers as those for a pion. If we now put
PA:
= fnm:r-
(12.24)
then (12.25) Here ~ ( xis )the pion field operator which creates r+ or destroys r-. Equation (24) is called the PCAC hypothesis. We note from Eq. (23), that in the limit, m: 3 0, t,he axial vector current is conserved. This implies that strong interactions have an approximate symmetry which is exact in the limit of zero pion mass. Such a symmetry is called chiral symmetry. Chiral symmetry manifests itself in the existence of massless pseudoscalar mesons called Nambu-Goldstone bosons. We shall come to this point again later. Here we discuss one of the important consequences of PCAC. We apply PCAC to neutron P-decay. Fkom Eq. (11.50b), we have
406
Properties of Weak Hadronic Currents and Chiral Symmetry
We note that pion pole contributes to the form factor f A ( q z ) only. It, does not contribute to gA(q2) nor h A ( q 2 ) . Separating out the pion pole contribution, we write
(12.27) where fA(q2) is the remaining part, of fA(q2) . &om Eqs. (26) and ( 2 7 ) , we get
Now if we assume t8hatin the limit, m: is conserved, we get, 2mNgA
(q2) - f i g r N N f r
3
0, the axial vector current
+ q2 f A ( q 2 ) = 0.
(12.29)
At q2 = 0, this gives (12.30) This is called the Goldberger-Treiman (G-T) relation. Thus G-T relation is exact in the chiral symmetry limit when pion mass is zero and the axial vector current, is conserved. This relation can be easily tested as all the quantities in Eq. (30) are experimentally known. This relation is valid within 6% agreement, with experiment. On the other hand, we note that
Partially Conserved Axial Vector Current Hypothesis (PCAC)
407
Using PCAC, viz. Eq. (24), we get
(12.32) Evaluating it at q2 = 0, m$ # 0, we again get the G-T relation. We conclude that the success of the G-T relation implies that deviations from chiral symmetry or equivalently from PCAC are indeed small. Finally, using SU(3) we can write for q2 M 0 for an octet of baryons [cf. Eq. (11.53)].
(Bk(P’) IAix I Bj ( P I )
(12.33) In particular for neutron &decay, we get
(12.34)
We define a four-vector sx = .li(p)yx75u(p).
(12.36)
p . s = o , s2=-1.
(12.37)
We note that
The vector sx thus gives the spin of the proton. To see it explicitly we go to the rest frame of the proton. In ,this frame, we get from Eq. (37), SO = 0, s2 = 1. From Eq. (36), we get
s = x +ax.
(12.38)
408
Properties of Weak Hadronic Currents and Chiral Symmetry
In quark model, we can write the axial-vector current Ai, = ~ T ~ T ~ + ( We define the quantity Aq as ( 2 r l 3 F ( P (4^/xYsQI P> = &Sx.
(12.39)
In particular for A ~ =x $ ( ~ ~ Y ~ ” I ~ &yx75d), ’L we have
1 ( 2 ~ ) ~ (Pp o( A 3 X ( p=) ~ ( A-UAd)sx rn
SO
(12.40)
that
nu-
Ad = Q A
=F+
D.
(12.41)
12.4 Current Algebra and Chiral Symmetry Isospin conservation implies that st,rong interactions are invariant, under SU(2) group generated by the charges:
I i ( t )= / I 4 , ( x , t ) d 3 2 , i
=
1,2,3.
(12.42)
In the same way we can define the axial charges
I f ( t )=
/ Aio(x,
t)d32, i = 1,2,3.
(12.43)
The generators of the isospin group SU(2) satisfy the commutation relations [Iz( t ) ,lj ( t ) ]= Z E i j k I k ( t ) . (12.44) Since I:(t)’s belong to the adjoint representation of SU(2) group, we have [Ii@),I$)] = i E & ( t ) . (12.45) We obtain a closed algebraic system by requiring that,
[I,”@), q t ) ]= i&Zj/Jk(t).
(12.46)
The last relation constitutes a major theoretical assumption. The commutation relations (44)-(46) represent the algebra of the group
Current Algebra and Chiral Symmetry
409
SU(2)xSU(2) generated by the vector and axial vector charges. This group is called the chiral SU(2) group. Let us now write the part of the QCD Lagrangian [cf. Eq. (7.32)] which involves u and d quarks:
where q =
( 1)
is an isodoublet field and we have suppressed
color indices. For mu = md this Lagrangian is invariant under the isospin transformation 4 u9, (12.48) +
where U is a special unitary matrix, exp [i:Ai] , Ai being constant. The associated vector current Ep = Q$ypq is conserved. The existence of nearly degenerate isospin multiplets of hadrons shows clearly that [mu- mdl is small compared to hadron mass scale (-1 GeV). Setting m, = md = m, we can write
where we have split q into “left-handed” and “right-handed” comp onents 1T 7 5 qL,R = 2 4. It is clear that in the limit m = 0, the Lagrangian (49) would be invariant under independent ‘chiral’ isospin transformations on q L and qR: qL
= U L q L , qR
-+
URqR
and not only Kp but also the axial vector current qypy5?q would be conserved. w e note that the mass term m ( q L q R 4- q R q L ) or in general the coupling to scalar and pseudoscalar fields
410
Properties of Weak Hadronic Currents and Chiral Symmetry
would break chiral symmetry. This also demonstrates that the forces between the quarks have to be vector in nature [mediated by spin 1 gluons, cf. the term ~T,,X Gpqin Eq. (47) or Eq. (49)]. As we shall see later mu 5 MeV, m d -10 MeV (these are called current quark masses, not to be confused with constituent quark masses of order 300 MeV [cf. Chap. 61) are small compared to the hadron scale of O(1 GeV) so that chiral symmetry is nearly exact,. Now if Aix were conserved, the axial charge 1; would commute with the Hamiltonian: N
[1,5,H]= 0.
(12.50)
Hence if we define
1:
1x1)= ieijk
I&)
1
(12.51)
use of Eq. (50) would imply that the states I&) are degenerate in mass with IX,)even though they have opposite parity. This is because 1 : has negative parity. This condition can be realized in either of the two ways:
1. The Wigner-Weyl realization of SU(2) symmetry, in which case l Y k ) would consist of “parity doublets” of IX,)e.g. if IXj) were pseudoscalar mesons, l Y k ) would be scalar mesons degenerate in mass with the pseudoscalar mesons. This is not what occurs in nature and therefore chiral symmetry is not, realized in nature in this way in contrast to the ordinary isospin symmetry which is realized in this way. 2. Spontaneously broken symmetry realization of SU(2), in which case l Y k ) would consist of IXj) plus an odd number of pions with vanishing four-momentum (called soft pions), the pion being a massless “Nambu-Goldstone” boson. In particular (12.52)
the first part being valid only for single-pion transitions, while
1i 10) = 0.
(12.53)
Current Algebra and Chiral Symmetry
41 1
+
As we shall see m: would involve (mu md)/2 as a factor and so a measure of explicit chiral symmetry breaking is provided by rn:/m: x 0.03, p being the non-strange (non Nambu-Goldstone) boson next to pion. The notion of (approximate) spontaneously broken chiral symmetry has been found useful in hadron physics and has given rise to many predictions involving soft pions which are in good agreement with the data [see bibliography]. One such prediction is the Goldberger-Treiman relation (30) : (12.54) to be compared with the experimental value 0.06 f 0.01 of the left-hand side. The above considerations can be easily generalized to SU(3). Thus the QCD Lagrangian (7.32) shows an approximate global symmetry in the limit mp 0, this Lagrangian is invariant, under the group SU(3) xSU(3) generated by the charges associated with the weak currents Jip. Thus the generators of the group are (i = 1 , . * * ,8). --$
Fi
=
J
F5
=
J ~ i o ( xt ,) d 3 2 .
V , ~ ( Xt)d3a: ,
They satisfy the commutation relations
[E,F j ]
= ZfijkFk
(12.55)
[F,,F,5]
= ZfijkFf
(12.56)
= ZfijkFk.
(12.57)
[F5,F.f]
The commutation relations (55) and (56) follow from flavor SU(3), the commutation relation (57) is a new assumption. Equivalently if we define 1 1 (12.58) F,L = - (& - F:) , :?I = - (Fi+)'?I 2 2
412
Properties of Weak Hadronic Currents and Chiral Symmetry
we get
Symmetry generated by the above group is called the chiral symmetry. If (R1,Rz) is a multiplet of group SU(3) xSU(3), then under parity (12.60) (Rl, Rz) (R21R1). -+
For example ( 8 , l ) + (1,8),(3,3*) + (3*,3). This means that if this symmetry is realized as a classification symmetry, we must have parity doublets. This is not the case in nature. No parity doublets are found. This implies that, the chiral symmetry is realized in the Nambu-Goldstone mode that is to say, there are eight, bosons which in the chiral limit have zero mass. As we have already seen, pions are the Nambii-Goldstone bosons which in the chiral SU(2) xSU(2) limit are massless. The eight, pseudoscalar mesons are identified with Nambu-Goldstone bosons of chiral group. The algebra generated by Fi and E5is called the chiral dgebra. This algebra has rather rich physical content because generators of the symmetry group can be identified with observables. The matrix elements can be measured in electxoweak interactions. This in fact provides evidence for chiral symmetry [see bibliography]. 12.4.1 Explicit breaking of chiral s y m m e t r y As already seen the chiral symmetry is spontaneously broken [cf. Eq. (52)]. Another way of expressing it, is that
(0 1441 0)
# 0 =+ K5 10) # 0.
(12.61)
To see this, we note that, in the quark model, we have the following commutation relations:
[F:, 5’31
= id,jkPk
i = 0, 1, ...,8
Current Algebra and Chiral Symmetry
413
(12.62)
where Xi si = q-q, 2
Xi
Pa = 4-754 (12.63) 2 are respectively the scalar and pseudoscalar densities. We note from Eqs. (62) that
I +
(0 [PI
2P2, F:-,]
2 2 10) = 22& (0 IS01O)+ (0 iz 0) . (12.64)
Now we expect that flavor SU(3) is realized in the usual way and is not spontaneously broken [cf. Eq. (53)]. This implies that
as
Thus, if
F4fi5 10) = 0.
(12.6513)
(so)o = (Gu + Jd + ss)o # 0
(12.66)
then we have from Eq. (64):
the condition for spontaneously broken symmetry [cf. Eq. (52)]. Let us write
(uu)o=
(q0{ =
S - S ) ~=
-.(say).
(12.68)
Hence we have the result that, (SO)*# 0 which implies that, chiral symmetry is spontaneously broken and (Sg),, = 0 implying that flavor SU(3) is not spontaneously broken.
414
Properties of Weak Hadronic Currents and Chiral Symmetry
We can write the QCD Hamiltonian density [cf. Eq. (7.32)]
as
3-1
=
= 3
+ (m,uu + mddd + m,ss) 2 3-10 + 6 ( 2 m + ms)S0+ - ( m - m,) ss + (m, - r n d ) fi
s 3
(12.69)
3-10+3-1’.
The Hamiltonian density 3-10 is chiral invariant,. Here f i +md). NOW
=
(1/2)(m,
(12.70) where
H ( t )= J’d3x71(tlx). The (charge) continuity equation d F5
2 =
dt
=
/d3x
(aAio(t’ at
+ V . & ( t , X)
/d3xPAi,
(12.71)
then converts Eq. (71) into
PAix
= -2
[F:13-1’].
(12.72)
E%om Eq. (72), we have (12.73) Using Eq. (52)) namely
(12.74)
Current Algebra and Chiral Symmetry
415
we obtain
f7r
-i
[ ( ~ j l J d X A , / 0 ) +(OJa’Ai~lrj)] = -i(OI[F’, [ F , f , % ’ ] ] I O ) .
2Jz The use of PCAC relation a X A = is symmetric in i and j ] .
(
fT/&)
[q, [c5,%‘I]
(12.75) m?.lri,then gives [m?j
10)
(12.76)
where 3-1‘ [cf. Eq. (69)] is
Substituting Eq. (77) into Eq. (76) and using Eqs. (68) and (62), one obtains
(12.78) Let A be the electromagnetic contribution due to photon exchange Since T + ) K+ form a U-spin multiplet the electromagnetic to contribution to mg* is also A while it, is zero for m:o, mgo, so that adding A in Eq. (78) for r+,K + , we get
mi*.
mi,
(12.79)
416
Properties of Weak Hadronic Currents and Chiral Symmetry
Here we have used the explicit breaking of chiral symmetry in calculating the current quark mass ratios in terms of masses of pseudoscalar mesons. When quark masses go t,o zero pseudoscalar mesons become zero mass Nambu-Goldstone bosons required by spontaneously broken chiral symmetry. 12.4.2 An application of chiral symmetry to non-leptonic decays of hyperons Consider the matrix elements [where B, and B, are members of the same baryon octet]:
(Bs(P’)
I pi?,HW] 1
B T
(PI) = (Bs(P’) ( p h v - HWF,S( Bv (PI)
(12.80) where i = 1,2,3. Using Eq. (74) and its hermitian conjugate, we can write it, as
(12.81) In other words in the limit qp = ( p - P ’ ) ~ 0 [called the soft, l ( p ) ) are pion limit], if the matrix elements ( B , (p’) r i ( q ) l H ~ B, non singular, then Eq. (81) gives ---f
+
Now Hw = Hz;;”H r [cf. Chap. 111 and it can be shown that for s-waves [HZ;;”], the amplitude on the left hand side of Eq. (82) is non-singular [see below] and we have
417
Current Algebra and Chiral Symmetry
Figure 1 Pole diagram in hyperon decay.
For pwaves [ H F ] ,one can apply the result (82) to
= -iJ”
f?r
I
I
(B”(p’) [F:, H F ] B, ( p ) )
(12.84)
where the Born terms are shown in Fig. 1. These are singular for Hz;;” in the limit qp 0 where mB = mk as they behave like 1/ I r n ~- mbI but for H F they behave like 1/ ( m+.m&I ~ and are non-singular. Now as we have seen in Chap. 11 [cf. Eq. (11.28)], the [AS1 = 1 non-leptonic Hamiltonian is ---f
Hw
GF
= -sin BCcos BC [Syp(1
fi
+ y5)21][Uyp(l+ y5)d] .
(12.85)
This being the product of two left, handed currents [FR= F, satisfy [F?,Hw] = 0
+Ff]
(12.86)
418
Properties of Weak Hadronic Currents and Chiral Symmetry
Figure 2 Triangle diagram for ro-+ 27 decay.
Furthermore F, (being the generator of SU(3) flavor group) acting onlB, > or IB, > produces a member of the same octet. To illustrate this point, consider for example, IB, >= / A > and < B,( =< pl , i = m. Then fi F1+i2
111) = 0 and (nl = (PI
F1+i2.
Thus for s-wave from Eqs. (83) and (86) (12.87) Also as shown in Chap. 11, in the exact SU(3) limit ( B , IHg"l BT)= 0. Thus the p-wave non-leptonic decays are given by the Born terms which are also determined by (B, IHE"l B,) as far as weak vertices are concerned. These were the results which we employed in Sec. 3.3~ of Chap. 11.
12.5 Axial Anomaly As seen in Chap. 7, 7ro -+ 27 is given by the triangle graph of Fig. 2. In the chiral limit (mu= md = 0), this triangle graph gives a finite value for the 7ro -+ 27 amplitude:
M(r0
-+
27) = E ~ * ( J E ~ ) E ~ * ( I C ~ ) E ~ , ,(12.88) ~~C~~C~
Axial Anomaly
419
with
(12.89) where N , is the number of colors, e. g. 3, e, = 2/3, ed = -1/3 while the Goldberger-Trieman relation for ( Q I A ~ ~ with I Q ) A3p = 51 (Wpysu. - &&) gives ( f l r / l / Z ) g l r q q= m, so that Eq. (88) gives
(12.90) It is important to remark that the result (90) is unaltered by radiative corrections to the quark triangle and Eq. (90) is independent of the masses of fermions in the loop. Equation (90) gives
(12.91) which is remarkably close to experiment with only 2% PCAC correction to the amplitude. The above result is often stated in terms of contribution to the amplitude due to an axial-vector “anomalous” divergence:
(12.92) where Fpu= Opa, - auup [upbeing electromagnetic potential] and w - &puapFap. 2 Note that Eq. (92) does not arise from equations of motion (72). That is why it is called “anomalous” divergence. Combining Eqs. (72) and (92), we have
(12.93) The first term on the right-hand side of Eq. (93) vanishes in the chiral limit but it is not so for the second term. The PCAC relation x becomes for A ~ thus
(12.94)
420
Properties of Weak Hadronic Currents and Chiral Symmetry
The “anomalous” divergence equations for
?78 and
70 are
a!
d X A k= ~ -S~FPyFPy,
(12.95)
47r
where lc = 8 or 0 and
2
+ e i + e q ] = 2z,
(12.96)
Similar considerations show that in QCD, t,he flavor SU(3) singlet current,
has “anomalous” divergence (12.97) where
GP’ = -1E P @ I
2
GmP
(12.98)
and G,, involving gluon field has been defined in Chap. 7 [cf. Eq. (7.31c)l. Thus
dXAox =
8
+
+
.CPu.
[m,.z~i7~u.mddiy5d+ m,siyss]
J:: --GPu
(12.99)
It is clear from Eq. (99) that the SU(3) singlet current is not conserved in chiral SU(3)@SU(3)limit. An application of this will be considered in Chap. 14.
QCD Sum Rules
421
12.6 QCD Sum Rules We have seen in Chap. 7 that the asymptotic freedom property of QCD makes it possible to calculate processes at short, distances or for large q2, q2 being the square of the momentum transfer. On the other hand, bound states of quarks and gluons (hadrons or hadron resonances) arise because of large distance confinement effects, i.e. strong coupling effects, which cannot be treated in perturbation theory. The idea of QCD sum rules is to calculate resonance parameters (masses, width) in terms of QCD parameters (as, quarks masses and number of other matrix elements which are introduced to parametrize the non-perturbative effects). We have also seen previously that in the absence of quark masses, the QCD Lagrangian shows a global chiral symmetry i.e. it is invariant under a global s U ~ ( 3x) s U ~ ( 3group. ) But this chiral symmetry is spontaneously broken i.e. the ground state is not invariant under this symmetry. This gives rise to [q = u , d , s] [cf. Eq. (Sl)]
(0 la41 0)
#0
leading to an octet of zero mass pseudoscalar mesons (so-called Nambu-Goldstone bosons; such bosons acquire masses when QCD Lagrangian is explicitly broken by the quark mass terms). The non-vanishing of the above quark condensate is a non-perturbative effect and gives rise to power corrections to asymptotic freedom effect, which is logarithmic. The essential point of the QCD sum rules i.e. to relate QCD and non-perturbative parameters of the above type with resonance parameters, is illustrated by the simplest of sum rules i.e. for a two-point function:
A (q2) = 1 IT
s-q2
i
The left-hand side is saturated with resonance so that 1.h.s. =
C mi 9:- q2 a
(12.101)
422
Properties of Weak Hadronic Currents and Chiral Symmetry
where (gi,mi) are resonance parameters. The right-hand side is useful only for large q 2 in which limit the perturbative QCD allows us to calculate the coefficients Ci(q2) in the operator product, expansion. In practice we want, to sahrate 1.h.s. by a few low lying resonances. Thus we should use some weighting factor to suppress large s contributions on 1.h.s. This is done by using Bore1 transform of the sum rule, which introduces a weighting factor involving a mass parameter M 2 ,which should be sufficiently large to suppress non leading terms on r.h.s. of Eq. (101) but not too large in order to suppress contribution from higher hadron states on 1.h.s. Thus the problem in practice reduces to finding a region of stability point, for M 2so that a small variation in M 2will not affect the physical parameters. In this way from QCD sum rules for two-point and three-point functions, a large number of constraints on hadron spectrum have been obtained providing not only a consistency check but also a useful phenomenological information on resonance as well as QCD parameters and on (Olqq10). For details see the bibliography.
Bibliography
12.7
423
Bibliography
1. R. E. Marshak, Riazuddin and C. P. Ryan, Theory of Weak Interaction in Particle Physics, Wiley-Interscience (1969). 2. E. Commins and P. H. Bucksbaum, Weak Interactions of Leptons and Quarks, Cambridge University Press, Cambridge, England (1983). 3. H. Georgi, Weak Interactions and Modern Particle Theory, Benjamin/Cummings, New York (1984). 4. T. D. Lee, Particle Physics and Introduction to Field Theory, Harwood Academic (revised edition 1988). For current algebra and chiral symmetry, in addition to the above, see 5. S. L. Adler and R. F. Dashan, Current Algebra and Application to Particle Physics, Benjamin, New York (1968). 6. S. B. 'Ikieman, R. Jackiw and D. J. Gross, Lectures on Current Algebra and its Applications, Princeton University Press, Princeton, New Jersey (1972). 7. V. de Alfaro, S. Fubini, G. F'urlan and C. Rossetti, Current in Hadron Physics, North Holland, Amsterdam (1973). 8. M. D. Scadron, Current Algebra, PCAC and the Quark Model, Rep. Prog. Physics, 44, 213 (1981). 9. C. H. Llewellyn Smith, Particle Phenomenology: The Standard Model, Proc. of the 1989 Scottish Universities Summer School, Physics of the Early Universe, OUTD-90-160. 10. J. F. Donoghue, Light Quark Masses and Chiral Symmetry, Ann. Rev. Nucl. Part. Sci. 39, 1 (1989). 11. 3. F. Donoghue, Chiral Symmetry as an Experimental Science, CERN-TH. 5667/90, Lectures presented at International School of Low-Energy Antiproton, Erice, Jan. 1990. For QCD Sum Rules, see 12. M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 385 and 448 (1979). 13. L. J. Reinders, QCD Sum Rules, An Introduction and Some Applications, CERN-TH-3701 (1983): Lectures presented at the
424
Properties of Weak Hadronic Currents and Chiral Symmetry
23rd Cracrow School of Theoretical Physics, Zakopane (1983). 14. S. Narison, QCD Spectral Sum Rules, World Scientific Lecture Notes in Physics-Vol. 26, World Scientific, Singapore (1990).
Chapter 13 ELECTROWEAK UNIFICATION 13.1 Introduction The Fermi theory of ,&decay cannot be the fundamental theory of weak interactions. It leads to many difficulties; it is non renormalizable theory. In this theory the scattering cross section for the process up e- -+ v, p- is given by Eq. (2.155):
+
+
O8
=
G; S .
(13.1)
IT
The above scattering is purely S-wave. Now Eq. (3.118) [XI = A2 = f1/2] gives ua = f 12F0I2[the factor 2 in the denominator is average over initial electron spin], where Fo = 7)o . Now the = f. maximum absorption occurs when qo = 0, so that IF0I2 5 Thus the partial wave unitarity gives
'::-'
5
(13.2) so that from Eq. (1)
G;
5
-s
7r
87r
S
or GFS
2Jz
51.
(13.3)
7r
Hence Fermi theory breaks down for s > ( ~ & / G F ) = (0.9 TeV)2. Therefore, we need a cut-off AF signifying new physics beyond AF 425
Electroweak Unification
426
where from Eq. (3)
A:wu
5 0.9 TeV.
(13.4)
Here PWU signifies that this has been obtained from partial wave unitarity. On the other hand if weak interactions are mediated through vector boson W , then instead of Eq. (l),we get
(2)(s+rnb)rnk 2
0,
=
32
T
s
(13.5)
which is finite for all energies, approaching the limiting value
Thus we see from Eq. (1) that the W-boson mass mw provides the cut-off AF. As we shall see mw w 80 GeV, so that mw << AF = 0.9 TeV . The charged weak interactions like electromagnetic interaction are vector in character (V - A ) and if the mediators of these interactions are vector bosons, then the universality of weak interactions suggests that the underlying theory of these interactions is a gauge theory. Since weak interactions have short range, the vector bosons associated with them must be massive. But the mass term is not gauge invariant. However, if the gauge symmet,ry is spontaneously broken, then the gauge vector bosons acquire mass. In this way all the desirable features of a gauge theory like universality and renormalizability are preserved.
13.2
Spontaneous Gauge Symmetry Breaking
Before discussing the gauge theory of weak interactions, we consider a simple model to illustrate the idea of spontaneous symmetry
Spontaneous Gauge Symmetry Breaking
427
breaking. Consider a simple Lagrangian
where
are left handed and right handed fermion fields respectively. 4 is a complex scalar field interacting with fermion having a coupling strength h. V($)is given by (13.8) Consider the gauge transformations
(13.9a)
Obviously the Lagrangian (6) is invariant, under the gauge transformations (9) if A1 and A2 are constants. The gauge group corresponding to gauge transformations (9) is U( l ) @ U ( l ) . If we require the Lagrangian (6) to be local gauge invariant, then we must
Electroweak Unification
428
intxodiice two massless gauge fields A, and BIL,which transform as 1 c 1
-8, A1
A,
+
A,-
B,
-+
B,+-a,
A2.
9
(13.10a)
(13.lob)
Then we can write the gauge invariant Lagrangian, by replacing by the covariant, derivat,ives:
8,
in Eq. (6). Hence the gauge invariant Lagrangian is given by 1
L = --A’”” A,, 4 -h
-
+
1
(a,+ ieA, - ZgB,) Q L Q + L G ) R i ~ ’ (a,+ ieA, + igB,) Q R
-B’” B,, 4
( ~ L ~ Q ~R R ~
+ +L
+ (ap+ 2 i g W ) 4 (a,- 2igB,)
27,
4 - V (4).
(13.11)
In Eq. (8), it, is usual t,o choose A > 0, since V (4) would have no minimum if X < 0. If in V (&), p2 > 0, then we have the ordinary scalar particles of mass p and V (4) has a local minimum at, q!J = 0. Then t8hemodel is not. interest,ing. However, if p2 < 0, then V (4) has a local minimum a t 1412 = -$ or 4 = This is shown in Fig. 1. By convention, we select, the positive sign. This is a classical approximation to the vacuum expectation value of q!J
&a.
(13.12) Although the Lagrangian (11) is invariant, tinder the local gauge transformations (9) and (lo), the non-vanishing expectation
Spontaneous Gauge Symmetry Breaking
Figure 1 Effective potential V
429
( 4 ) for p2 < 0, showing local minima.
value of 4 means that the gauge symmetry is broken i.e. the vacuum or the ground state is not invariant under the gauge transformation (9b). This can be seen as follows. From Eq. (9b)
U-' (Az) $U (Az) = e 2iA2 4,
(13.13)
Therefore,
(0 IU-'
(A2)
4U (Az)lO)
= e2iA2(0 1410) .
(13.14)
If the vacuum is gauge invariant, then
u (A21 10) = 10)
(13.15)
(0 1-610) = e2iA2(0 1410 ) .
(13.16)
and Eq. (14) gives
Thus if (0 I q5 10) # 0, then e2aA2 = 1 for any A2, a contradiction. Hence U(A2) 10) # lo), and the gauge symmetry is spontaneously broken i.e. U1 x U2 -+ U1, UIis unbroken.
Electroweak Unification
430
We now show, how spontaneous symmetry breaking leads to the massive vector boson B p . It, is convenient to define a new field 4’;
(4‘)
0,
=
(13.17)
where $1 and $2 are hermitian fields with zero expectation values. The Lagrangian (11), in terms of the fields $1 and q52 has the form
L
=
1 4
--A
”
1 4
A,, - -B’” B,,
+ -21 (4g21j2) B’B,
-
2gvBL”d,$2
F’rom the Lagrangian (18), we derive some interesting results. If the gauge group U2 is a global gauge group, then we do not require the vector boson B, and from the Lagrangian (18), we see that the scalar field 41 has acquired a mass &%?, the scalar (pseudo) field 4 2 is massless, the fermion field Q has also acquired a mass$ and the vector field A, corresponding to unbroken gauge symmetry U1 is massless. Hence we have the Goldstone-Nambu theorem. A spontaneous breakdown of global symmetry leads to a massless scalar particle. But when U2 is a local gauge symmetry, then due to the presence of the term 2 g 1 ) Bpa,q$2 a straightforward interpretation of (18) is not possible. But we can eliminate this term by a field dependent gauge transformation. Actually what happens is that 13, $2 combines with B, (which has only transverse
Spontaneous Gauge Symmetry Breaking
431
components) to form a single massive spin 1 field, 8, 4 2 now becomes longitudinal mode of spin 1 field. This can explicitly be seen as follows: Choose the gauge function A2(x) to be ?$. Then under the gauge transformations
(13.19) the Lagrangian (18) becomes (removing ^):
1
L = --A 4
p’
1 APv - 4B””B,, Q
h -
--QQp
Jz
+ -21 (aJ
+ 21 (49’11~)BpB,
- eGyWA,
- g$y,y5Q!B,
1 - - (21129 p2 2
(13.20)
It is clear from Eq. (20), that, the would be Goldstone boson field q(z) has been transformed away; it has been eaten away by the field B, to give a longitudinal component. This mechanism is called the Higgs-Kibble mechanism. The massive scalar particle p is called the Higgs particle. To summarize: (1) No massless scalar boson appears. (2) A, which is associated with unbroken gauge symmetry (electric charge conservation) has zero mass. (3) The vector boson B, has acquired a mass m B = 2gv. ( 4 ) The fermion field has acquired a mass rnf = ( 5 ) Both the masses of B, and \k arise due to the same symmetry breaking mechanism. (6) A massive scalar particle with mass&%? appears. This particle is called Higgs particle. Presence of Higgs scalar is an essential feature of spontaneously broken gauge symmetry.
3.
432
Electroweak Unification
13.3 Renormalizability We give here few remarks about the renormalizability of a gauge theory. Now the fields A, and B, cannot be determined uniquely by field equations. In order to quantize these fields, one has to fix a gauge that is to say break gauge invariance. For the photon field A,, a term added to the Lagrangian for this purpose is Photon propagator is then givcri by
--$[-’
For the field B,, the gauge fixing term is
It, is so chosen that, it cancels awkward looking mixing term B”8,42 in the Lagrangian (18). [ is a parameter which determines the gauge. The propagator for the vector boson B, is given by
The field
42 has its propagator i k2 -
r2 rng
These form of propagators are expected to give a renormalizable t,heory for any finite value of [ since they have good high k2 behavior, falling like $. This is called R-gauge. The fields B, and 4 2 separately have no physical significance. In particular the poles at, k2 = t2m i are tinphysical and are canceled out in any S-matrix 00, the element,, which is also independent of [, In the limit, [ B-meson propagator becomes --f
and 4 2 propagator vanishes. This is called unitary ( U ) gauge. The renormalizability is not obvious in this gauge.
433
Electroweak Unification
13.4 Electroweak Unification As we have discussed in Chap. 11, the leptonic charged current ~. of weak interactions has the form V.7, (1 - 7 s ) e = 2 i i , ~ ~ ’ eThe corresponding hadronic charged weak current can be writken as %yp (1 - 7 5 ) d‘ = 2fi~y,di. Here d’ means that it is not, mass eigenstate. This suggests that we consider
as left handed doublets in a weak isospin space. The weak currents are then associated with weak isospin raising and lowering operators
(13.21) where $ L is any of the above doublets, r+ = (71 +ir2) and 7- = (rl - 2 7 2 ) . Let, the charges associated with these currents be Q+ and Q- . These charges generate an s U ~ ( 2 algebra )
[Q+, Q-I
= 2Q3.
(13.22)
The current associated with the charge Q3 is given by
Jl
G L 1~ T ~ ~ , Q L .
(13.23)
) The gauge transformation corresponding to the group s U ~ ( 2 is
qL(z) -,
q L
(a
(z) = exp i- . A ( r )
)
@L
(4.
(13.24)
Then the Lagrangian
W,” I
(13.25)
Electroweak Unification
434
where
D,
3,
=z
+ 2 g21- r .
(w
W, = 8,
+ igW,,
(13.26)
--r.W,) 1
'"-2
W,,
=
W,,
a,W, - a,W, - gW, x W, 1 - T . W,, = D,W, - D,W,
(13.27a)
+ 29 [W,, WU]
(13.2713)
2
= a,wu - &WP
9
is invariant under the gauge transformations: QL
(4
w, where
uQ L ( 4
-+
-+
2 uw,ut--ua,ut
(13.28a)
9
U is given in Eq.(24). For A infinit,esimal, we get QL
(x) W,
+ 22 7
A (x)) Q L (x)
-+
(1
-+
W,-AxW,--a,A.
9
1
(13.28b)
9
The gauge group s U ~ ( 2 leads ) to a neutral current, J," which is neither observed experimentally nor is identical with the electromagnetic current. It, is possible to unify weak and electromagnetic forces into a single gauge force, if we extend the gauge group to SUL(2) x U y (1).For this group we have two gauge couplings g and g' associated with SUL(2) and Uy(1) respectively. The weak hypercharge Y is defined by the relation Q = t 3 i Y = i ~ 3iY.The gauge vector bosons W*, W obelong to the adjoint, representation of SUL(2) and vector boson B, is associated with Uy(1). Fermions belong to either fundamental representation [doublet] or trivial representation [singlet]. The structure of charged
+
+
435
Electroweak Unification
weak currents suggest the following assignments: 1st generation dR
Y
-1
-2
413
113
-213
2nd generation CR
,
3rd generation
(
: ) L 1
In order to brez the gauge symme :y spontaneousl: so that weak vector bosons acquire their mass, we need a Higgs doublet $:
$=($).
Y=l.
(13.29)
The Lagrangian invariant under the local gauge transformations QL
3
e x p ( i r . A + - Yi ~ A o 2
QR
is given by
3
exp ( i ~ f iQh R ~)
(13.30)
Electroweak Unification
436
where W,, is given in Eq. (27a) and
(13.34)
a = 1 , 2 with q R l = e R or d R , ~ R =Z u ~Under . the infinitesimal gauge transformations (30), vector fields W, transform as given in Eq. (28), but B, transforms as
B,
+ B, -
1
-8 Ao. 9’
(13.35)
cL
In order to break the gauge symmetry spontaneously, assume that,
(13.36)
J-rL2/x,
where ZI = ($)o = (0 1410). In this way, not, only SUL(2) is broken but, Uy(1) is also broken, but it leaves the group V(1) corresponding to electric charge unbroken viz. s U ~ ( 2x) & ( I ) is broken to U Q ( ~ )We . can now write Eq. (29) as
(13.37) where 4+ and hermitian fields 41 and 4 2 have zero vacuum expectation values. We can select a gauge such that g5+ and b2 disappear
437
Electroweak Unification
ap+&
from the theory. Instead and a p + 2 provide longitudinal components to W* and one of neutral vector bosons respectively. Thus out, of the four gauge vector bosons, three become massive and the remaining one remains massless. This massless vector boson is the photon corresponding to unbroken UQ(1) symmetry. All this amounts to replacing given in Eq. (37) by (41 = H )
+
+=(
K$) 0
(13.38)
With Eq. (38), the following term of the Lagrangian (31)
gives LW-H
=
+g2 ( H 2 + 27)H + u2) (2WlWp- + W 3 p w3.) 8 +-9’2 ( H 2 + 2vH + B”B, 8 -& ( H 2 + 27)H + W”B,, (13.39) 4 -1a p H a p H 2
712)
71’)
where WE = (WlpfiW2p)/&. Fkom this equation, it is clear that vector bosons W’, have acquired a mass:
rnb = 29 1 2 v 2.
(13.40a)
For the neutral vect,or bosons, the mass terms in Eq. (39) give the matrix (13.40b) Since det (Ad2)= 0, therefore one of the eigenvalues of M 2 is zero. The mass matrix (40b) can be diagonalized by defining the physical
438
Electroweak Unification
fields A,, 2,
A,
=
2,
=
cos BwB, -sinOwB,
+ sin OW W3,, + cosOwW3,.
(13.41)
Then we get 2
A,: photon
mA = 0,
(13.42)
(13.43) where 9’ tanow = -
(13.44)
9
and the parameter p-
mL m i C O S ~ew
=
1.
(13.45)
The fermion masses are given by m . - h.2 -
1)
%fi
(13.46)
and Higgs boson mass is given by 2 mH = 2Xv 2 = -2p2,
(13.47)
From Eq. (31), using Eqs. (41), (44), (40a) and (46), the Lagrangian for the fermions can be written as: LF
=
Gi 27,
(9
--
8, - m i- -H ) 8i 2mW
9 -2 7’1(1 - 75) (T+W;
+ T-w;) 92
2 f i
-e
G iy p Qi Q iA,
+ 2 cos 8w
*i
7’ (gvi - Y 5 9 A i ) ~i
(13.48)
Electroweak Unification
where
9i =
, d:
439
( 7; ) ( ::) , and
= VZj dj
and
Qi
e = 9’ cosOw = g sinow =
(V : CKM matrix),
is its charge. g v i and
gvi =
(q3- 2Q3
Tf
1 2
= -Tf,
mi
Q A ~ are
sin 8,)
is the mass of ith
given by
,gAi
= q3
1 2
773 = - 7 3
(13.49a) (13.4913)
We note that the interaction part of the Lagrangian can be written as
Lint = -gsinOw J,”, A, - -(J” 2fi
W’,
+ h.c.) - ___ J z p 2, cos 8w (13.50a)
where
1
= - J3’ - sin2OW J,”, 2
-4sin2 OW ( - E 7p e
+...
2 + -fi 3
7’ u -
(13.50d)
Electroweak Unification
440
where ellipses in Eqs. (50) indicate repitition for tlhe second and third generations. For low momentiim transfer phenomena, q2 << mb, m i , we can write (13.51)
p = 1;
m2,
=
mw
=
&e2
-
7ra
~ G sin2 F OW f i G sin2 ~ OW 37.3 GeV 2 37.3 GeV sin2OW
mz=-- mw cosew
-
74’6 GeV 2 74.6 GeV. sin2Bw
(13.52) (13.53)
(13.54)
Note that, p = 1 is a consequence of the fact that Higgs scalar 4 is an SUL(2) doublet. The effective neutral current coupling [see Eq. (49)] is g2
g2
GF
8pz.
(13.55) m$ cos2Ow = p- mb = Finally, we note that for t,he Higgs vacuum expectation value I ) , using Eqs. (39) and (51), we get 2
1)
=-f
i
~
- (246 GeV)2
(13.56)
F
This gives the weak interaction scale i.e. the energy scale after which the weak interactions become as strong as electromagnetic interaction. The fermion masses are given by
(13.57a) h: = 2 f i G F m ; i.e. the Yukawa couplings are very weak.
(13.5713)
Electroweak Unification
44 1
We conclude this section with the following remarks:
1. A definite prediction of electroweak unification is the existence of weak neutral current J,” with the same effective coupling as charged currents J: . This current, has been found experimentally.
2. The existence of vector bosons W*, 2, with definite masses given in Eqs. (53) and (54). 3. The theory has one free parameter sin2Bw.
At low energies q2 <<
rnk, one test of the model is to de-
termine sin28w from different classes of experiments. If sin2&, comes out to be the same in all these experiments, it will support, the model. The true test of the model is the existence of vector bosons. This requires much higher energies. We first discuss low energy consequences of the electroweak unification. The vector bosons W* and 2 have been found experimentally with masses predicted by the model. 13.4.1 Experimental consequences of the electroweak unification Low energy phenomena q2 << rnb : From the Lagrangian (50), for low momentum transfer phenomena [q2 << rnk, m i ] we can write the effective Lagrangians for charged and neutral currents:
(13.58) (13.59) It is convenient, to write J,”:
J:
= J:
(v)
+ J f ( e ) + J;
(h),
(13.60)
where
(13.61)
Electroweak Unification
442
Table 13.1
Since the net strangeness of the proton is zero, we will assume that strange quark s and heavy flavor quarks c, b etc. make negligible contribution to J,”(h) for proton and neutron targets. Then we can write the effective Lagrangians for various neutral current processes as follows:
(13.64)
From Eqs. (50), we can determine the parameters E L ( e ) , & R ( e ) , EL(^), & ~ ( iCli, ) , and Czi(e), (i = u , d ) . They are given in Table 1.
Electroweak Unification
443
13.4.2 Need for radiative corrections
Before we discuss the experiments in support, of the standard model, let us summarize here the three parameters (not counting Higgs meson mass m H and the fermion masses) which the minimal model (with p = 1) has: (a) fine structure constant a = 1/137.0359895(61) determined from the Josephson effect (b) the Fermi coupling conGeV-2 determined from the muon stant GF = 1.166389(22)x life-time {including lepton mass and O(0)radiative corrections [cf. Eq. (11.40c)]}. (c) sin’ Ow,determined from neutral current processes or the W and 2 masses. Now a best fit to the neutral current neutrino reactions data gives sin2Ow = 0.2255 f 0.0021.
(13.67)
This implies that, the theory wit,hout, radiative corrections gives through the relations [cf. Eqs. (52) and (54)] /sin2
ew
=
4
sin’ ew
~
(13.68a) (13.6813)
where 112
A0
=
(L) = (37.2802 GeV) ,
JZGF mw = 78.42 GeV,
mz
= 89.14
GeV.
(13.69) (13.70)
These values are to be compared with the experimental ones mw = 80.39f0.06 GeV and mz = 91.1867f0.002 GeV. This shows a need for radiative corrections. First we note that the t,wo coupling contants g and 9’ which determine the strength of weak interactions are related to e through e = g g ’ / ( g 2 g ‘ 2 ) . Since most measurements are made at 2 peak, therefore most, convenient mass scale for these couplings is at m z . Thus one should take into consideration the running of QED coupling countant [see Appendix B] which
+
Electroweak Unification
444
gives the value of a at, mz :
where f = e, p, 7, u,d , s, c and b and N,f = 3 for quarks and 1 for leptons. Equation (71) can be directly evaluated for leptons, since their masses are well known. For the light hadronic part, quark masses are not available a s reasonable input parameters. The 5flavor contribution to IIrr is extracted from the experimental data on e+e- 3 hadrons. The best, estimate leads to , . 1 a (mz) = 128.88 f 0.09 na = I---- a = 0.0595 & 0.0007. (13.72) a (mz) Thus knowing G F , a (mz) and mz,one should be in a position to predict, all electroweak observables, including the mixing angle sin2Ow = <. However, the lowest, order relation mw / mz = cos Ow 9 and some other lowest, order relations are affected by the fermion loops in gauge bosons propagaters. Thus the relation (68b) is mod* ified to
mi$ mi cos2Ow
=p =
(1
+Ap).
(13.73)
The leading contribution to A p comes from the top quark loop (mb<< mt>to the self energies of W and 2 bosons (see Fig. 2): Ap=-
3GF mp 87r2fi
= 0.0096 f 0.006
(13.74)
for mt= 175 f 5 GeV. By contrast, the Higgs boson contribution to Ap at, the one loop level is logarithmic:
Electroweak Unification
445
Figure 2 Top and bottom quarks loop contribution t o W and 2 boson self energies.
Radiative corrections are scheme dependent,, leading to sin2 Ow values which differ by small factors which depend on mt and mH. A useful scheme [called the on-shell scheme] is to take tree level formula sin2&, = 1 - m& / m; as the definition of renormalized sin20w to all orders in perturbation theory i.e. sin20w t s k = I-m&/m2,. Now the tree level expression (69) is modified to
while using Eq. (73),
=
sin2Ow
-
Apcos2 Ow.
(13.76)
Thus the tree level expressions (68) are modified to
(1
-
$ ) m ~=mk
s k = m2, sw 2 cw 2 =
~
A:
1 - Ar
(13.77)
Electroweak Unification
446
where
where (Ar)remainder contains all possible contributions riot, included in the fermionic contributions Aa and Ap. Likewise, taking into account radiative corrections, the relationship between the W*-boson mass, s k and Gg in the standard model gets modified
A2
2
mw= where A2 = A;*
ff
2
sw (1 - Arw)
= A 0 12- A a1' so
(1 - Ar)
=
(13.79)
that from Eqs. (77)-(79)
(1 - Aa) (1 - A r w ) ,
(13.80)
(13.81) where we have used Eq. (69) and have defined (13.82) The relation (79) defines Arw, which is completely determined by purely weak correction, once Q (mz) is specified. For L E P physics, sin2 6w is usually defined from 2 t pt p- effective vert,ex. At, the tree level we have [cf. Table 11 2
--+
f f :2 cos9 ow f 7 P ( g t - 9 i
75)
f
Electroweak Unification
447
The weak (non Q E D ) connections to the 2 can be conveniently written as
--+
f
f effective vertex
where Pf = 1-ap,Ap being given in Eq. (74) and [cf. Eq. (76)] 2 sw =
s;
&/g$
=
Sf2
- Cf-2
*P
1-Ap
s k +&
= 1 - 4Sf2
Ap.
(13.85)
(13.86) (13.87)
The radiative correction functions A r w , (pf - 1) become the basis on which the corrected standard model and experiments are confronted. These functions involve, among other parameters, the top quark mass mt whose value due to quadratic dependence of some of these functions on mt is numerically important and the Higgs boson mass which enters only through logarithm and hence is not effectively bounded. Now if we use the L E P value s; = 0.23189 f 0.00024 for leptons, we can obtain s& = 0.2245 f 0.0010 from Eq. (85), with Ap given in Eq. (74) and hence mw through the relation (rnz = 91.187) s b = l-m&/m; : m w = 80.30f0.05 GeV which is consistent with the direct vector boson mass measurements: mw = 80.39 f 0.06 GeV and the standard model best fit value: m w = 80.372 GeV obtained from Eq. (77) (including higher order terms) from m z , G F , o and mt, mH. Finally including mt, m w from the direct experimental measurements, together with s& from neutrino scattering, global fits of
448
Figure 3
Electroweak Unification
v,-electron
scattering through 2-boson exchange.
the standard model paraniet,ers to elect,roweak precision data give
mt = 171.1 f 4 . 9 GeV
a, ( m ~ = ) 0.119 f 0.003 GeV. The upper limit, on m H at, the 95 % CL is the theoretical uncertainty is included.
m H
(13.88)
< 262 GeV, where
13.4.3 Ezperiments which determine sin2& We now discuss three sets of experiments to determine sin20w. 1. Consider the process
This process can occur only through 2 exchange (Fig. 3). The effective Lagrangian for this process is given by Eq. (64). For this case the laboratory cross-section for E, >> me give
where the upper (lower) sign refers to v, (D,) , E, is the incident energy and Gg rn,/27~ = 4.31 x 10-42cm2/GeV. The expressions for gF and g$ in terms of sin2 Ow are given in Table 1. The most
Electroweak Unification
449
accurate leptonic measurement,s of sin2f3w are from the ratio
+ 16sin4OW 1 - 4sin2Ow + 16sin4OW
R = -crUpe - - 3 - 12 sin’ OW UPpe
(13.90) ’
The most precise experiment (Charm 11) determined not only sin2Bw but g $ , ~as well. The experimental results are
f 0.017 -0.503 f 0.017 0.2326 f 0.0084.
9;
= -0.035
9:
=
sin2Ow
=
(13.91)
2. The deep inelastic neutrino scattering v,+N -+ v,+X (isoscalar target), gives a precise determination of sin2Ow on the mass shell i.e. The relevant Lagrangian is given in Eq. (65). The ratio R, E cr,”,”/c~,”, of neutral to charged cross-section has been measured to 1% accuracy. A simple zeroth order approximation gives
sk.
R,
= 9:
+ 9;
T,
RD= 92
+ 9;
/ T,
(13.92a)
where
92 =
EL
+
(u)2
EL
1 ( d ) 2N“ 2 - sin2
+ -95 sin4ew
and T oEN / is the ratio of V and u charged current crosssections, which can be measured directly. In parton model T M E ) / (1 , where E M 0.125 is the ratio of the fraction of t e nucleon’s momentum carried by antiquarks to that carried by quarks. Now from Eq. (92) on using Eq. (73) and (76) we can write R, = -1 - S ~ 2+ ( ~ + T ) - S5W4+ S L 2 9 (13.93a)
+
+ BE)
(13.93b)
Electroweak Unification
450
R,
- rRD=
(1 - T )
[21 - s b (1 + Ap)]
(13.94)
It is clear from Eq. (93a), that, dependence of R, on Ap is weak. Hence this equation is useful to determine s$. Using the experimental values R, = 0.317 f 0.003, T = 0.440, and A p = 0.0096 we obtain from Eq. (93a) &, = 0.2242 f 0.0022, and (with rnz = 91.187) rnw = 80.32 f 0.11 GeV. The recent value quoted for s& from vN scattering is 0.2255 f 0.9021, which gives mw = 80.25 d~ 0.11 GeV fully consistent with the directly measured value for mw = 80.39 2r 0.06 GeV. We note from Eq. (94) that if we plot R, versus RDit, gives a straight line with a slope determined by T . This provides an acciirate method to determine p s$ from the experimental data. 3. Parity violating deep inelastic eD scattering: The relevant Lagrangian for this process through 2 exchange, which is parity violating, is given in Eq. (66). There is an interference with the parity conserving process through photon exchange. This gives us the parity-violating asymmetry (13.95) where oR,L is the cross-section for the deep inelastic scattering of a right (left)-handed electron e R , L N + e X . In the quark parton model (see Chap.14 for this model)
A
-=
q2
a1
+
1 - (1 - y)z a2
1
+ (1 - y ) 2 ’
(13.96a)
where q2 < 0 is the momentum transfer and this essentially comes through the photon propagator which appears in the photon exchange process. Here y is the fractional energy transfer from the electron to hadrons. For the deuteron or other isoscalar target neglecting the s quark and antiquarks,
Decay Widths of W and Z Bosons
Figure 4
451
W-boson decay.
(13.9613) where we have used Table 1 in the second step of these formulae. The experimental values for a1 and a2 can be used to determine the mixing angle sin2Ow.
13.5 Decay Widths of W and 2 Bosons
+
Consider the decay W - --t efie shown in Fig. 4: From Eqs. (49) and (50b), the decay amplitude F is given by
F = -u-9 - ( k 1 )-/A (1 - 75) ZI (k2) 2 f i
*
EX,
(13.97a)
where E X is the polarization of W-boson. From Eq. (97), the decay width can be easily calculated and is given by [ in the limit when we neglect the lepton masses as compared with rnw] :
(13.9713) We can also calculate the hadronic decays of W - from the basic processes like W ti d, C s. Again we get an expression like
452
Electroweak Unification
+ F)
(97b), except that we multiply it by a factor N , = 3 (1 M 3.12, where the factor 3 is due to color and the factor in the parentheses is a QCD correction. In this case we also neglect quark masses as compared with W-mass. This is a good approximation with the exception of r-quark which channel is not, open as mt = 175 f5 GeV. Since in weak interactions a linear combination of mass eigenstates d , s and b enters, therefore, we have to multiply the decay rates by square of such factors as IVud12, I&s12, (VuSl2, etc. [see Sec.13.101. The link of one generation to succeeding generations is very weak, therefore, we will put IVUdl2x cos28, M I, I V , ~MI ~cos20, M 1, I V , , ~M~ sin20, M 0, M sin48, M 0. Hence the relative decay widths for three generations are given by:
(v,~I'
Thus we get
F (W+ ---t I+v,)
M
227.5 f 0.3 MeV,
F (W+-+u,dl)
M
(708 f 1) MeV
rF
M
2.098 GeV,
(13.98)
to be compared with the experimental value 2.097 f0.003 GeV for pot W'
+ f,the decay amplitude F is given
For the decay 2 t f by [from Eqs. (49) and ( ~ O C ) ]
(13.99) and the decay width is given by
(13.100)
453
Decay Widths of W and Z Bosom
where N,f = 1 or 3 (1 + a,/.rr) for f = lepton (I) or quark q . First we note that gAf = -12 ' gvf = (-1/2 IQfl 2sin2&). In the presence of radiative corrections, we have [cf. Eqs. (84) , (86) and (8711
+
Thus we get
(13.102b) It, is convenient to write s; = (1
+ Ah) s;
(13.103)
where Ak signifies non QED corrections and si is given in Eq. (82) and has the value 0.2311 for mZ = 91.187 GeV. Then
1- 4 IQfl(l+ Ak) s : ) ~ ] . (13.104) Let us write
ro(z-+
I">
ro(z+ qq)
= =
% fi!.2%4 [I~ + (1 - 4si)2]
32
[I
+ (1 - 4 [Qql
s ; ) ~ ]3 (1
(13.105a)
+ ")IT
.
(13.105b)
Electroweak Unification
454
From Eqs. (105), we get on using si = 0.2311,
ro( 2 -+ vv) ro(Z e-e+) -+
= =
165.9 MeV ro(Z-+ p p +
(13.106a) =
ro z
-+ 7 - T + )
83.4 (83.91 f 0.10) MeV (13.106b) ro(2-+ U U ) = r0(2--+ CC) = 296.9 MeV (13.106~) ro -+ dz) = ro( Z ---+ SS) = ro( Z -+ b6) = 382.6 MeV. =
(z
(13.106d) Thus
r0(2
hadrons) = 1742 MeV = 1.742 (1.7432 f 0.0023) GeV (13.107a) ro ( 2 -+ invisible) = r z - ( r h a d 3re+,-) = (2.494) - (1.742 0.250) ---f
+
+
= 0.502 GeV = 502 MeV
(13.107b)
where experimentally r z = 2.4939 f0.0024 GeV. The values given in parenthesis are experimental values. Now 3rO ( 2 3 v D ) = 498 MeV; hence one concludes from Eq. (107b) that#N , = 3 i.e. there are three generations of neutrinos or three generations of fermions. It may be noted that in calculating the decay widths we have put fermion mass as zero. This is a very good approximation; the decay width for Z -+ bb may need some improvement if mbis not, neglected Even the theoretical values as given by the Born approximation ro are not bad, the small discrepancy can be explained by taking into account the radiative corrections given in Eq. (106). We can also use Eq. (104) to constrain Ak and Ap by using the experimental value for r (Z-+ e+ e-) . Another important observable is forward-backward asymmetry measured at LEP. Consider the process e-e+ --t ff as depicted in Fig. 5 . The cross-section for e-e+ -+ ff can be
Decay Widths of W and Z Bosom
Figure 5
455
Production of 2 in e-e+ collision and its decay into
ff
pair.
easily calculated by using Eq. (99). It is given by (in the limit s >> 4~712, 4m;, p e M 1, pj 1)
da -- -{(1+cos28) dN,f dSl
[Q;-21
4s
Qf llevf 6 sin2OW cos28w Rex (4
+ u:)(vp + a;) I x ( 4 I?] + (16(v,2sin2 Ow cos2 Ow)? +case
[- 16sin24QfOw cos2Ow R e x (s)
8vevjaeaf + (16 sin2 Ow cos2 Ow)2
(13.108a)
where S
x(4 ve
vf
(13.108b)
= s-m2,+imzrz' = 2gve = -1 4sin2ow, a, = 2g.4, = -I f f = 2gvf = 2T3, - 4Qf sin2Ow, af = 2 g A f = 2T3,. (13.108~)
+
Near and on the peak, integrated cross section is dominated by 2-exchange and we get from Eq. (100):
Electroweak Unification
456
where (13.109b) and the effect, of radiative corrections are contained in 6(s), the large effects due to initial e* bremsstrahliing are represented in 6. The other radiative corrections which lead to improved Born approximation have already been discussed in Secs. 3.2 and 4. The LEP data is fitted with an additional modification i.e. by replacing s - m i i m z r z by s - m i i&rz . Note that the expression for l?j is given in Eq. (100). Thus by measuring cr:eak for a particular final state e.g. e+e- itself, one can directly obtain r e e / r ~or reer z rffand therefore r j j . These widths have already been discussed in the beginning of this section. The forward-backward asymmetry is defined as:
+
+
'
(13.110) It is clear that this asymmetry is given by cos 0 term in Eq. (108a). Near and on the 2-peak, we get, from Eqs. (110) and (108c)
(13.112) Taking into account the radiative corrections [cf. Eqs. (101) and (103)], we get, from Eq. (112) 3 (1 - 4.S;)l
[1+ (1 - 4 s g 3(1-4(l+Ak)~%)~ [l 4 (1 Ak) s?]' '
+
+
(13.113)
Decay Widths of W and Z Bosons
457
where s: = 0.23116 and
AkBo -
-do(+) _---
3 (1 - 4s2,)2 [I
= 0.01685 (0.01683 Az 0.00096) .
+ (1 - ~ s Z ) ~ ]
do(-)- 2a2
1
S
(13.115b) Hence the polarization at s = m i is given by
- --2vf af2 ' -- -2 gvf/gAf . 1 + s;f/sif up af
+
(13.116)
We can also write Eq. (116) in terms of effective mixing angle s f : 1- 4s;
Al= 2 1
+ (1 - 4s?)2'
( 13.117)
Using the value A, = 0.1431 f 0.0046 we obtain s; = 0.23201 f 0.00057 to be compared with si = 0.23116 f 0.00022.
Electroweak Unification
458
13.6 Tests of Yang-Mills Character of Gauge Bosons The vector bosons self-couplings are given by the Lagrangian (31)
Lw
1 4
= --
p,w, - 8”W, - g (W, x W,)I2.
( 13.118a)
This gives the trilinear W+W-W3 coupling as
Lw
.9
= 2-
2
[(a,w3v - avW3,)
( W - w + ”- W+Pw-”)
+ (a,w,+- avw;) ( W 3 W ” - W3W-p ” > -
(ay;
-
a,w-P > (W 3 W + ”- W.”W+”)]. (13.118b)
Using W; = sin &A, + cos O,Z,,, the above equation gives for the W+W-y and W+W-Z vertices
+ ( h- k 2 ) ,
-
(h- kz)pga?] ,
(13.118c)
p and y are the indices of polarization vectors of W - , W+,W 3respectively. On the other hand from Eq. (39), t,he Higgs
where a ,
coupling to gauge bosons is given by
LW-H
g2
= -(H
8
+ 1 1 ) ~2W:W-’ + cos21 ow Z,Zp] ~
(13.119a)
and the Yiikawa coupling of Higgs to leptons is given by
LlrH where
Jz
hl = -rnl 2)
=
1 -hll lH,
(13.119b)
a
112
= ( 2 2 / 2 G ~ ) rnl.
(13.120)
Tests of Yang-Mills Character of Gauge Bosons
459
One process in which the trilinear couplings can be tested directly is e+ e- -+ W+ W - .
+
+
In the lowest order of 9, the diagrams shown in Fig. 6 contribute to this process. We are interested in the high energy behavior of the amplitude M . The bad behavior comes from the longit,udinal polarization of W’s. For this case p = 0. The longitudinal polarization vector E: for a W-boson of four-momentum kp is given by
(13.121) gives the It is the first term in Eq. (121) viz. %which mW worst high energy behavior. The amplitude may grow with high energy due to this term, if it is not compensated. In fact 51s E + 00 the diagrams of Fig. 6 give
(13.122) (13.123)
It is clear from Eqs. (122) and (123) that there is no possibility of cancellation between MLL( u ) and MLL( b ) even if e = g sin Ow. The third diagram, arises due to trilinear couplings - a feature of gauge theory. All the three diagrams cancel the bad high energy
460
Electroweak Unification
Figure 6 Production of W-W+ pair in e-e+ collision through v,,y and 2 boson exchange.
Tests of Yang-Mills Character of Gauge Bosons
46 1
I I I
:H I
I I
I I
Figure 7 Production of W-W+ pair in e-e+ collision through Higgs boson exchange.
behavior except for the last term in Eq. (122), which gives S-wave cross-section for s >> m&,os = This is in conflict with the unitarity constraint [Eq. (2)] os 5 This conflict thus starts at,
2s.
F.
4 4
s = -7r
2
= (1.2 TeV)
.
(13.125)
GF
However, even this term is canceled by the diagram (Fig. 7) due to Higgs exchange. This is because this diagram gives for s >> m&: (13.126) Thus there is no trouble with the high energy behavior in the standard model if m$ < (1.2 TeV)2. There is similar cancellation for the amplitude MLT, which for each individual diagram goes as constant when s t 00. o(efe- ---f W - W’) depends crucially on gauge cancellation discussed above. For example, o(Y - exchange) 1rCX2S for m& 96sin4(JwmL , this would be the only contribution
Electroweak Unification
462
Behavior of ( r s M with energy E .
Figure 8
without W - W+y and W - W+Z vertices. On the other hand, with the above cancellation (13.127)
The cross section also contains the threshold factor \il which tends to 1 as s 4 00. Thus the cross section grows near the threshold and then falls like at large values of fi >> rnw. The cross section is 10-35~m2 at its maximum which occurs at about 40 GeV above W+W- threshold. The situation is shown in Fig. 8.
5
N
Upper Bound
463
13.7 Higgs Boson Mass
The Higgs potential
v (4) = p 2 d 2 +
q52 = $4
(13.128)
goes over to
1
1
1
- -Xu4 V ( H ) = (2Av2) H 2 + AvH4 - -AH4 4 4
(13.129a)
when the symmetry is spontaneously broken; p2 = -Xu2 (A > 0) . Thus we see that the Higgs boson maSs (13.129b) is arbitrary. We now discuss theoretical bounds on the Higgs boson mass. 13.8
Upper Bound
(a) Unitarity: We have seen in Sec. 13.6 that the Higgs boson contribution to the cross section for the process e-
+ e+
---f
Wz
+ WL
is given by (13.130) Then comparing it with Eq. (2), we get (13.131) This requires a “cut-off” (signaling new physics beyond As,):
I ( 4 f i ? ~ / G F ) ’ ’= ~ (1.2 TeV)
(13.132a)
Electroweak Unification
464
To avoid this conflict, the Higgs mass m~ should be such that ULH
< Ag,Wci = (1.2 TeV) .
(13.132b)
We saw at, the beginning of this chapter that, m w << A$Eu. Whether similar thing happens or not for m H only experiments will tell. (b) Finiteness of couplings: The Higgs-self coupling X is not asymptotically free. In X 44 theories, the renormalized group equation gives (see appendix) d
d In q2 ((I2)=
(13.133)
(q2)
This gives
(13.134) The minus sign in this equation indicates that, the Higgs coupling is not asymptotically free. In fact, it, implies that regardless of how small X ( u 2 ) is, X ( q 2 ) will eventually blow up a t some large energy scale q = A. In order to avoid this and to guarantee positivity of x (A) : A (A) < m, x (112) < giving rn?{ : [ I t 2 = f i G F
&+ ’ n 7
A]
The upper bound on m H is related logarithmically to the scale A up to which the standard model is assumed to be valid. For some values of A, the upper bound on mH is given below
wi 10’ GeV 10’’ GeV
159 GeV
I 144 GeV I
Thus we see that if we assume the standard model to be valid up to Planck scale , then
m H
5 144 GeV.
Higgs Boson Searches
465
We note that the top quark Yukawa coupling ht = fimt/v, for mt = 175 GeV, can be of order 1. Top quark coupling modifies the renormalization group equation for the Higgs boson coupling A. Top loop corrections reduce X for increasing top-Yukawa coilpling. Such an effect, is shown in Fig. 9. We also note that nonperturbative effects as the quartic Higgs coupling becomes large have also been estimated, mostly in the context of the lattice-Higgs model. Again a cut off on the parameter X (u)provides an upper bound on mH : m H < 700 GeV. Finally the precision electroweak data give as previously noted m~ = 76
-47
GeV.
(13.136)
13.9 Higgs Boson Searches The search for Higgs is one of the objectives of new accelerators. The dominant production mechanism in LEP2, is e-e+ ---t 2 -+ Z H . The tree level cross section is given by
Here the standard model couplings [cf. Eqs.(ll3) and (126)]
have been used in deriving Eq. (137). X is the phase space factor
(13.139) With the present L E P energies a lower limit has been established on Higgs mass m H 2 98 GeV. (13.140)
Electroweak Unification
466
700 500 n
3
300
100
70
50
1.OO
125
150
175 200
225 250
mt [GeVI Figure 9 Bounds on the mass of the Higgs boson in the SM. Here A denotes the energy scale a t which the Higgs boson system of the SM would become strongly interacting (upper bound); The lower bound follows from the requirement of vacuum stability. [9]
Higgs Boson Searches
467
LEP2 running at energies 200 GeV should enable the Higgs to be discovered if mH 5 110 GeV. We now discuss the decays of Higgs, since Higgs searches involve its decay widths. For H -+ ff, the coupling involved is
h2f -- 2m; 7 -
- 2&m;Gp,
(13.141)
giving (13.142a) where
N,f = 1 =3 =
f = 1* f = q
for for
(1 - 4m;
/
ma) 112 .
(13.142b)
It may be noted that there are important QCD corrections for H --f qq. The bulk of QCD radiative corrections can be mapped into the scale dependence of the quark mass, evaluated at the Higgs mass i.e. use mf at mH i.e. mf (mH) in Eq. (142) [see Eq. (B.47)]. For H 4 W+W- and 22, the couplings are given by [cf. Eq.(1191 (13.143a) where
4 = 3. These give the widths 8mw (13.143b)
(13.1434
Electroweak Unification
468
where
(13.144) It is useful to remember that for mH
r ( H -+
M
1.4 TeV,
1
V V ) M -m$GF 2
N
(13.145)
m H .
To sum up the standard model is in very good shape, but Higgs boson H is still a missing link.
13.10
GIM Mechanism
Since in weak interactions the flavor quantum numbers are not conserved, weak interaction eigenstates of different generations, d', s' and b' are not identical with mass eigenstates d, s and b. These states are linear combinations of d , s and b. Thus we can write
d'
=
Vud
d
+ V,, s + Vub b
Kdd+K,s+Kbb bt = I& d + V , , s + & b b.
s'
=
(13.146)
The quarks of one generation are linked to those of the succeeding generations wit,h decreasing strength. Thus for example V,b << V,, < This is illustrated by the following diagram [Fig. 101: If we confine ourselves to ordinary and strange hadrons, then we can safely put Vub = 0, but we cannot ignore Vc,,since charmed quark is linked to strange quark with maximum strength. As a first approximation, we can ignore the third generation = sine,, as given by completely and can put Vud = C O S ~ , , Cabibbo theory. Thus we can write d' = d cos 8, s sin 6,. In the weak neutral current, we have a term of the form
v&.
v,,
d" rfi(1 - 75) d'
+
GIM Mechanism
469
U
S
Figure 10 Relative strengths of flavor changing transitions. = cos28,
2 yp (1- 75) d + sin28,
+sin8,cosOC [d yp (1 - y5) s
3 yp ( I - y5) s
+ s yp (1 - y5)d] (13.147)
which arises from the doublet
( ).
The above term can give
rise to the following processes (Figs. l l a , l l b ) . It is clear from Figs. 11 that both the processes
K+ K+
--+
-,
.rr++v+v T O + e + +v,
occur with equal strength. But experimentally
i.e. the strangeness changing neutral current is very much suppressed compared with the strangeness changing charged current. Here the charmed quark c comes to the rescue. If we put V,, = - sin 8, and V,, = cos O,, then s' = -d sin 0, s cos 8, and we get a
+
Electroweak Unification
470
+ +
Figure 11 Decay K+ -+ T+ u fi through neutral current and T” e+ u, through charged current.
K+
---$
+ +
term
from the doublet
( z, ) .
From Eqs. (147) and (148), it is clear that,
strangeness changing terms are canceled and J,” does not contain any strangeness changing term. This mechanism to eliminate the strangeness changing neutral current in tree approximation was suggested by Glashow, Iliapoulas and Maiani (GIM) before the experimental discovery of charm. The A S = 2, K” -+ transition shown in Fig. 12 is second order in GI;..With GIM mechanism, a complete cancellation between u and c couplings occur if m, = m,. With the known experimental value for this transition, a limit, on the mass of rn, can be put and it was predicted that m, must be less than a few GeV and this is what, was found later experimentally.
GIM Mechanism
471
c
I
K?
Figure 12
Box diagrams for A S = 2, KO -'?I
lRO transitions.
Electroweak Unification
472
13.11
Cabibbo-Kobayashi-Maskawa Matrix
Three generations of fermions are linked with each other by weak interactions. The states d’, s’, and b’ are not mass eigenstates. They are related to mass eigenstates d, s and b as follows:
(+(
;)
(13.149)
where V is a 3 x 3 matrix:
(13.150) called ‘Cabibbo-Kobayashi-Maskawa’ (CMK) matrix. The hadronic charged weak current can be written as J,” ( h )= (3,c,
q r p (1 - 75) v
(3
(13.151)
and J j ( h ) which is a part of the neutral current:
We want weak neutral currents to be flavor diagonal as flavor changing neutral currents are very much suppressed. Hence we must have V+V = VV+ = 1, (13.153) i.e. V must be a unitary matrix. Thus this matrix has nine real parameters. These parameters are the same in number as unitary
473
CabibbeKobayashi-Maskawa Matrix
group write
U3.
Now
has three diagonal matrices, so that we can
U3
v = eiOAo
eiPXe
eiaX3
c
eicy’X3
e
@’As
(13.154)
,
where C is a 3 x 3 unitary matrix with 4 real parameters. The five parameters 8, a , p, a’ and p’ can be absorbed into redefinitions of phases of u , c , t and d, s, b quarks. Thus we can write
V = RzRiC’R3 where
(13.155)
R1,R2 and R3 are 3 x 3 rotation matrices: R1
=
(:I
1
d:
:),
0
1
R 2 = ( 01
c2 0
s:),
0 -s2
c2
0 (13.156)
and
c is a unitary matrix which can be written as (13.157)
Hence we have s1c3
s1s2
-c1s2c3
-~
s1s3 2
ei6 ~
3-clsqsg
+~
2
ei6 ~
3
(13.158) where ci = cos Bi,
si = sin Oi.
(13.159)
There is an arbitrary phase S, which makes the Lagrangian density non-real. Thus the Lagrangian density violates t,imereversal invariance. By CPT theorem, it violates C P invariance. Thus there is an attractive possibility of accommodating C P violation
Electroweak Unification
474
in 3 generation model; this cannot be done in two generation model [see chapter 151. If we consider t,he three generations, the fermion mass matrix for u,c, t and d , s, b quarks can be written as
+
[
( )] +
(zL,i+iL,bL) M
h.c.
(13.160)
Without any loss of generality, we can take Mu to be diagonal matrix viz.
Mu= ( m u
mc
J
*
(13.161)
It is clear from Eq. (160) that,
V+fGV= I&,
(13.162)
where Md is now diagonal matrix. Below we give the experimental values of CKM matrix elements:
Axial Anomaly
475
7"
Figure 13 Axial vector current anomaly from the triangle graph.
+
Note that IVUdI2 lVUSl2 + IVUb12is consistent,with 1 as required by unitarity.
13.12
Axial Anomaly
For a theory to be renormalizable, it is essential that vector and axid vector currents are conserved. In electroweak gauge theories, before spontaneous symmetry breaking, fermions are massless and it is, therefore, expected that axial vector current, is also conserved. But this is not so, in fact as seen in Chap.12, axial vector current receives anomalous contribution from the triangle graph: a closed fermion loop with one axial-vector vertex and two vector vertices as shown in Fig. 13. This anomalous contribution is equivalent to the statement that in the zero fermion mass limit, the divergence of axial vector current is given by (13.163)
where FPVis the field tensor of the vector field and g is the coupling constant as shown in Fig. 13. The contribution from A graph arises only if A graph is odd in axial couplings. This contribution is independent of fermion ma6ses and is unaltered by radiative corrections. In QED, such graphs do not cause any trouble as photon is not coupled to axial
Electroweak Unification
476
Figure 14
Process e-e+
-+
27 through A graph.
current,. Nor does it cause any problem if one or more of the currents is associated with a global symmetry of the theory. In such a case, it, can even be useful as for example the case for 7ro -+ 27, which arises due to the anomaly as discussed in Chap. 12. In electroweak theory, such graphs are not absent. For example, in the process e-e+ -+ yy shown in Fig. 14, the A graph can cause trouble as it would give bad high energy behavior. To ensure the renormalizability of electroweak t,heory, it, is, therefore, essential to ensure the cancellation of A anomalies. Consider a gauge group G, where the coupling of the fermions to gauge bosons is given by
Lint = G ~ i y ’(6)’” ”
+ igAtWap)‘J!L + GRiy’” (ap+ igAFWap)‘J!R. (13.164)
The current coupled to gauge bosons is given by
J;
1-
= 2 Q y p (1 - 75)A t Q
+ :Gyp 2
+7 5 ) Afq.
(1
(13.165)
Here A: and A: are hermitian matrices; they satisfy the following commutation relations
9 L and ‘J!R need not transform in the same way under G as is the in general. The A-anomaly case in electroweak group. A: #
At
Axial Anomaly
477
(being independent of fermion masses) is proportional to
where - sign arises since it is odd axial vector vertices which give anomaly and it has to be symmetric in two indices say a and b . Thus [ { } denotes anticommutator]
A L
= 2-7-
AZc
=
({A:l
T r ({A:,
(13.167a)
A:} A:) A:}
A:).
A:)
A:>
(13.167b)
Theory is thus anomaly free when
Tr
A t ) A,")
- Tr
({A:,
=0
(13.168)
for all values of a, b, c.
Examples (i) Vector or vector like gauge theory:
AL = AR # 0.
(13.169)
A,L = A:
(13.170)
Af; = U-lA,RU,
(13.171)
For such theories either or where
U is a fixed unitary matrix. The gauge current is given by JL
+
=
% L ~ , A ~ Q L% ~ y , h f Q ~ GLy,A:QL + @Ry,UAtU-lQ~
=
Gy,A,Q,
=
where 9 = @L
+u-l@~,
(13.172)
Aa =
At,
(13.173)
is a pure vector. Note that in general the redefinition of @ generates y5 terms in the fermion mass matrix. Such a theory is caIled vectorlike.
Electroweak Unification
478
In QCD, the left handed and right handed quarks belong to the fundamental representation 3 of SUc(3).Thus it, is a vector theory and is anomaly free. (ii) AL = AR = 0. In this case, fermion representation is such that anomalies cancel separately for left handed and right handed fermions. This is the case for example for S U ( 2 ) . For the fundamental representation 2 of S U ( 2 ) , A, = ir, and since Tbr, = 2&b,
+
The representation 2 is a real representation in S U ( 2 ) . But this is not, the case for S U ( n ) , n > 2, e.g. representation 3 of SU(3) is not equivalent to 3*. Thus S U ( n ) , n, > 2 is not safe in general. However, fermions belonging to an octet representation of SU(3) are anomaly free since octet representation is real. This can be seen as follows: If A, form a representation, -A: also form a representation. The negative sign arises, since matrices A: satisfy the commutation relation [A:, A:] = -i f a b c l \ z . (13.175) Hence -A: form a representation conjugate to A,. If (as in the case for real representation),
A, where
=
-U-lA:U,
(13.176)
U is a unitary matrix, t,heri
Thus in general real representations are safe. They do not produce axial anomaly. However, a safe representation need not be real. (iii) The standard model SUc(3) x SU(2) x U(1).
Axial Anomaly
479
We need to consider S U (2) x U (1) only as SUc(3) is anomaly free. The matrices Atand A: are given by
1 1 Q = -73+-Y. 2 2
(13.178)
Now T R ({Tf,
TF} 7:)
(13.179)
=
so that from Eq. (168), we have to show that
Tr ({&T~}YL) Tr YL, 2 6 a b Tr [2Q - 731
= 2bab =
= 4bab
Tr Q = 0
(13.180)
and
(13.181)
Tr [Y;] - T r [Yi]= 0 for the cancellation of anomalies. Now
( 13.182)
T r [Yi] = 8Tr [Q3] Tr [Y,"]= Tr [SO3 + 6Q 732 - 6Q2 73 - T:] .~
8TrQ3
=
+ 6TrQ - 6Tr (Q2
73)
(13.183)
But
Tr[Q3] Tr
[Q27-3]
0;
TrQ
0:
Tw3 = 0.
(13.184)
Hence for the cancellation of anomaly, we must have
Now
Tr Q = 0.
(13.185)
2 1 T r Q = [O - 1+ 3(- - -)] = 0. 3 3
(13.186)
480
Electroweak Unification
Hence in the standard model; lept,on anomalies cancel quark anomalies. Note that, in the cancellation of anomalies, color plays a crucial role. Left-handed fermions anomalies cancel among t,hemselves and so do the right-handed fermions anomalies.
Bibliography
13.13
48 1
Bibliography
1. J. C. Taylor, Gauge theories of weak interactions, Cambridge University Press, Cambridge, U. K. (1976). 2. M. A. Beg and A. Sirlin, Gauge theories of weak interactions, Ann. Rev. Nucl. Sci., 24, 379 (1974); Gauge theories of weak interactions 11, Phys. Rep. 88 C, 1 (1982). 3. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Meld Theory (Addision-Wesley, Reading, Mass. 1995). 4. G. Altarelli, “The Standard Electroweak Theory and Beyond” CERN-TH / 98-348 hepph / 9811456. 5. J. Ellis, “Beyond Standard Model for Hill walkers” CERN-TH / 98-329,hepph 9812235 6. J. L. Rosner, “New developments in precision electroweak physics” Comment Nucl. Phys. 22, 205 (1998). 7. W. Hollik, “Standard Model Theory” CERN-TH / 98-358; KATP-18-1998 hep-ph / 9811313, Plenary talk at the XXIX Int. Conf. HEP, Vancouver Canada (1998). 8. M. E. Peskin,“ Beyond standard Model” in proceedings of 1996 European School of High Energy Physics CERN 97-03, Eds: N. Ellis and M. Neubert. 9. M. Spirce and P. M Zerwar, “Electroweak Symmetry Breaking and Higgs Physics” CERL-TH / 97-379, DESY 97-261 hep-ph. / 9803257 10. Particle Data Group, The Euro. Phys. Journal 3, 1-4 (1998).
Chapter 14 DEEP INELASTIC SCATTERING 14.1 Introduction Lepton-nucleon scattering is an excellent tool to study the structure of nucleon. Electron’$nuon) scattering clearly shows that nucleon has a structure. Consider for example the scattering e+p+e‘+X.
Let E be the energy of the incident electron e and E‘ be the energy of the scattered electron. Let q = k - k’ be the momentum transfer. Then in the lab. frame, the four momenta P, k and Ic‘ of the target (proton), initial electron and the scattered electron are given by
P
f
k‘ E
(M,O), k (E’,k’).
=
(E,k)
Neglecting the mass of the lepton, we have
k - k’= EE’cosO.
(14.la)
We define another invariant v:
Mu=P.q. 483
(14.lb)
Deep Inelastic Scattering
484
In the lab. frame
u
= 40 =
( E - E').
(14.lc)
We also define the invariant mass: 2
s = px = (4
+ P ) 2= q2 + M 2 + 2Mv.
(14.ld)
Note that 2 M u + q2 2 0; for elastic Scattering 2Mv = -q2. The elastic scattering of electrons on spinless proton can be written in terms of Mott cross section:
da dR
(14.2a) Mott
where
(14.2b) The structure of the proton manifests itself in term of the form factor F ( q 2 ) .In elastic scattering proton recoils as a whole and the scattering is coherent. The form factor F ( q 2 ) measures the charge distribution of the proton, viz.
(14.3a) If we expand F ( q 2 ) in powers of q 2 , we get
F(0)
=
/ p ( r ) d 3 r= 1
ylq2=o =
-27r
(14.3b) ( r 2 )is called the mean square charge radius.
DeepInelastic Lepton-Nucleon Scattering
485
It is convenient to write the Mott cross section in the form
($),,,,
4Ta2
=
-
2e
E’
(7) E C O S 2
This is the scattering cross section for the scattering of electrons on spinless (structureless) particles of mass m. The scattering cross section for the scattering of electrons on structureless spin 1/2 particles can be calculated using the standard trace techniques and is given by [Q2 = -q2] 47m2 E‘ -da - - -cos2 dQ2 Q4 E
14.2 Deep-Inelastic Lepton-Nucleon Scattering We now consider the inelastic scattering of electrons on nucleons (see Fig. 1). For this case the matrix elements are
The cross-section is given by [cf. Chap. 21
x
m2
S(Px + k‘ - k - P ) 2 L , , W p ” ” , EE‘
where [see the Appendix A]
vin
lkl =-
E
(14.6b)
Deep Inelastic Scattering
486
Figure 1
Inelastic charged lepton-proton scattering.
(14.6~) and
Here S denotes the spin of the target and En denotes the sum over all the quantum numbers of state X and integration over d3Px. Then the differential cross-section is given by
Assuming invariance under C, P and T and conservation of the electromagnetic current = 0, the Lorentz structure of
487
DeepInelastic Lepton-Nucleon Scattering
Wp” is
Here S2 = SpSp = -1, S - P = 0 and FI and F2 are spin averaged structure functions: MWl E F1 and VWZ_= Fz while the remaining two are spin dependent structure functions. In Fig. 2, we show the plot of Q2(= -q2) versus 2 M v where we have defined the variables:
x=-
E-E’ 2Mv ’ y = E = - E Q2
Y
(14.8a)
OIyyl. Now
so that
2 P . q - Q 2 2 0 or 0 5
IC
5 1.
(14.8b)
If hadron masses are not important, F’s could not depend on
Q2and one might expect that scale invariance holds in the asymptotic (Bjorken) limit Q 2 ,v + 00 with x fixed. In the “naive” quark model (where the virtual photon interacts with point like constituents), in the limit of quark masses ---t 0, there are no dimensions and this suggests that in the asymptotic limit the structure
488
Deep Inelastic Scattering
Mu Figure 2 Plot of momentum transfer Q 2 versus energy transfer u = E - E’ in charged lepton-proton scattering, showing various kinematic regions.
489
DeepInelastic Lepton-Nucleon Scattering
functions scale:
Mwi(v, Q 2 ) vwz(v,Q 2 )
f
J'2(v,Q 2 )
91,2(v, Q 2 )
---t
91,2(z).
Fi(v,
Q2)
4
Fi(2)
+
F2(z)
(14.9)
In QCD, however, this scaling is broken but, only by logarithms of
Q2/A&m.
F'rom Eqs. (6) and (7), the spin averaged cross-section is
given by d2a dS2 dE'
Wz(v,Q 2 )
+ 2 tan'
0 2
1
-Wl(v, Q2) . (14.10a)
It is instructive to write this cross-section in the form d2a 9I WZ(V, Q2) 2 tan2 -Wl(v, Q')] . dQ2du 2 (14. lob)
+
We now define right and left, polarized cross-sections as UR,L
= (T &
A(T,
(14.11)
where d2a/dQ2dv is given in Eq. (10). In terms of the variables 2 z, y and K = (I - $)[= 1 - ? i!& +? 1 in the scaling limit and is Q2 a measure of how close one is to the limit, Q 2 -+ 001, we have d2a
- -
2(1 - y)
dx d y
1 + 2y2(K - 1)) F2]
(14.12) 2 given by and polarized asymmetry A a = CTR- a ~ / is
dAa
47ra2
- = dx dy
ME
Q4
x
[ {(':: + COSP
2 1.-
-
-(K-
)
1) g1 - y ( K - l)g2
11
(14.13)
490
Deep Inelastic Scattering
At high energies y + 0 and F2 and g1 dominate. It, may be noted that g2 has never been measured. In Eq. (13) p is the angle between k and spin quantization direction S. If the target is longitudinally polarized p = 0. The presence of the structure functions in Eq. (10) indicates that proton is not a point particle. The structure of t,he proton can be probed in two ways - one by elastic lepton-nucleon scattering and second by deep inelastic lepton-nucleon scattering. First we discuss the elastic scattering for which v = Q2/2M. For this case the structure functions are given by
where T = Q2/2M. Thus from Eq. (lo), we have
+ 27 tan2 2 [FI(Q2)+ F2(Q2)I2}.
(14.15)
The form factors for the proton are normalized to F,P(O) = 1, F l ( 0 ) = /cCp and for the neutron Fp(0) = 0, FT(0) = ten where IC* = 1.792 and K, = -1.913 are anomalous magnetic moments of the proton and the neutron respectively. Experimental data is analyzed in terms of Sachs form factors
These form factors are normalized as follows: GpE(0) = 1, GpiM(0) = pp = 2.792, Gk(0) = 0 and G&(O) = pn. In terms of GE and G M ,
DeepInelastic Lepton-Nucleon Scattering
49 1
t,he elastic scattering cross-section is given by
(14.17) The experimental data is fitted remarkably well by a single form factor
GnE(q2) = 0,
(14.18)
where rn; = 0.71 GeV2. From Eq. (15), we get [cf. Eq. (3b)l 12 (14.19) - 0.66fm2, (T$) n = 0.
(s)Mo
as Now Eqs. (14) and (15) clearly show that $$ -+ Q2 + 00 i.e. cross section rapidly falls as Q2 become large, clearly showing that the nucleon has a “diffused” structure in the elastic region. But the behavior of the structure functions W2 and W1 is quite different in the deep inelastic region. The experimental data in this re ion indicate that the cross section stays large and is of the characteristics of a point particle. This clearly order of indicates that in this region the scattering is incoherent, and is what one would expect if a nucleon consists of non-interacting or weakly interacting point like constituents called partons (quarks). This scattering region thus gives us information about the elementary constituents of nucleon, i.e. about their charges, spin and flavor. Moreover, the structure functions vW2 and MWl show Bjorken scaling i.e. vW2 and MWl -+ F ~ ( Zand ) F l ( z ) as Q2, v -+ 00 where z = is fixed. This is clearly indicated in Fig. 3 where Fz(z) is plotted against Q2 for various values of 2. The above characteristics lead to parton model of deep inelastic scattering which we now discuss.
qs)Mott,
492
Deep Iiielastic Scattering
Figure 3 The structure function Fz measured by the CERN muon experiments, ( a ) proton ( b ) nucleon in deuterium.
493
Parton Model
14.3 Parton Model Partons are quarks (spin 1/2), antiquarks (spin 1/2) and gluons (spin 1). Gluons do not, contribute here since they carry no electric charge. Thus we shall deal with spin 1/2 partons. If the target is a free quark of flavor i , of mass m and charge ei, we have from Eq. (64
+
x S4(p Q - pn).
(14.20a)
Now G h = d 4 p n 6 b i - m2]and we obtain from Eq. (20a) pno
m
--y 27r
r
m e 3 a ( p s ) y P[11+
x
S(2p . q - Q 2 ) .
4 + m]y v u ( p s ) (14.20b)
To proceed’further, we make use of t,he following identities of Dirac matrices algebra [see Appendix A],
Deep Inelastic Scattering
494
Thus the comparison with Eq. (7) gives
1 2 91i = -b(x 2 - l)ei,
g2i
(14.2213)
= 0.
Hence from Eqs. (12) and (13) for a spin 1/2 parton i,
(14.23a)
dAai dx d y
47ra2
- -e:mEcosp
-
Q4
1
1) 6(z - I).
(14.23b)
The comparison of Eq. (23) with Eqs. (12) and (13) clearly shows that if we replace 6(1 - x) in Eq. (23) by some distribution functions F ( x ) and g ( 2 ) we get Eqs. (12) and (13). Hence it follows that in the scaling region, the nucleon is behaving as if it consists of point-like constituents and the structure function Fzi(x)or Fli(2) or g l i ( 2 ) gives us the 2-distribution of point-like constituents inside the nucleon. The point-like constituents have been assumed to be free i.e. interaction between them can be neglected in the scaling region. This is compatible with QCD, as QCD is asymptotically free. More accurately one can write Fz and Fl as F2(x, Q 2 ) ,F l ( 2 , Q 2 ) ;but the dependence on Q2 is very weak (logarithmic). The following physical picture emerges. In the deep inelastic region, the virtual photon interacts in an incoherent manner and probes roughly the instantaneous construction of proton. In the center of mass frame of electron and proton, we can write (neglecting lepton mass):
k
= ( P , 0, 0, P ) , P =
[(AdZ+ P
y
21
(
P 1+ )";"
0, 0, -P
]
Parton Model
495
2Mu - Q2 4P Let us assume that the target (proton) has point-like constituents .~ called partons of flavor, i. Neglecting any parton momentxm transverse to the target, let us assume that the longitudinal momentum of a parton is given by p = x P . The time of interaction of photon is given by 1 4P 2P 7 = - = qo ~ M -vQ2 Mv(1 - X ) ' 40 =
~
-4
The energy of a parton =
M
z P ( 1 iph), 2 +m2 so
that the lifetime of virtual parton states is 1
2P For x not going to 0 or 1, T << T in the deep inelastic region so that one can consider the partons contained in the proton as free during the interaction. Hence in the deep inelastic region the photon interacts with the constituents of proton as depicted in Fig.
4. If the target is built from partons of type i and the probability for a parton i to have momentum fraction x' to x' + dx' is fi(x'), then p - q = x'P q = Mvx', -
Vparton -
P q - -vx M -' m m
1
, 1
) -S(X S(Q2 - 2p 4) = S(Q2 - ~ M v x ' = 2Mu *
- z')
and Eq. (22a) becomes
x
S(a:
- XI).
(14.24)
Deep Inelastic Scattering
496
P
/ /
hadrons
XP
Figure 4
Since dQ2dv d2u
0: W p ” ,
The parton model.
we should write
or, on using Eq. (24),
6(x - d ) f Z ( Z ’ ) d d .
(14.25)
Using then the expression (7) for MWpv/27r, it follows that in the parton model:
(14.26) while [cf. Eqs. (24) and (7)]
(14.27)
497
Parton Model
where T and denote respectively parton spin parallel and antiparallel to proton spin S. The relation F'(z) = 21cFI(z), which is a consequence of parton having spin f, is well satisfied experimentally. For the proton target [denoting fi(x)conveniently by q ( 2 ) +Q(z),ei -+ eq and with spin sum understood 1, we have from Eq. (26)
C
plP=
+
mi [q(z) ~ ( z ).]
(14.28a)
q=u, d, ...
In other words
(14.28b) Applying isospin conservation so that, the u ( d ) flavored parton distribution in the proton is the same as d(u) flavored parton distribution in the neutron, whilst, the s and c . ., distributions remain unchanged being isoscalar, the neutron structure function becomes: F,"" = Ic - d ( 2 ) 91 (?A(.) fi(2)) * . .
[:(
+ +)) +
+
+
.]
(14.28~)
In the above equations u(z),d ( z ) , are the probabilities that parton (antiparton) of flavor u, d , - - -carries , a fraction 2 of the momentum of the proton or t8heneutron. For an isosinglet, target, N , we get, F,e N ( 2 ) f -1 (FiP(2)+F;""(x)) 2 5 [u(.) G(z) d ( z ) = x - a s ,
{
+
+
+ 441
(14.28d)
-
means the contributions of other quarks like c , b, t. Note is just the average squared charge of the u, d quarks. We have thus seen that the parton model leads to the Bjorken scaling of the structure functions in the deep inelastic scattering.
Here that
&
Deep Inelastic Scattering
498
14.4
Deep Inelastic Neutrino-Nucleon Scattering
Let us consider the processes
De+N ve+N
+
--+
!++X !-+X
The matrix elements are given by
T’
=
2
(2&
x8(k)y,(l
/%
- y5)21(k’)
(X [PIP ) ,
x.ci(lc’)y,(l - y5)21(lc) ( X IPtIP ) ,
(14.29a)
(14.2913)
where J P = V , - A,. Thkn we have to replace in Eq. (6b) e4/q4 by G$/2 and L,, by
2 L”” = [k,k:
- g,,k k’k,kl f i~,,,gk*lc”] , (14.29~) memu while W,, now contains for the spin averaged case three structure functions Wl, W2 and W,. The third function W3 arises due to V - A interference term and appears in Eq. (7) as &~,’*@P~qpF3, with vW3 = F3. The cross-section is given by pu
d2 0D,Y
dQ2 dv
-
[
G$ 1 E’ 0 e --W . , ,(v,Q 2 )cos2 - + 2W:’”(v, Q2)sin2 2 nE 2 2 (14.30)
In order to discuss the scaling, we again express the cross-sections in terms of the variables IC and y. The st,riict,iirefunctions show the following scaling behavior in the deep inelastic region:
V W [ ’ ~ (QV2,) -+ F{’y(x) MW:’y(v, Q2) -+ F,”)”(z) V W ~ ’ ~Q(2V) ,+ F:”’(x).
(14.31)
Deep Inelastic Neutrino-Nucleon Scattering
499
The cross section can be written
+-Y2 2xF.”(z) 2
For the basic processes
we get for quarks by substituting Eqs. (22b) without e: and S(x - 1) in Eq. (32),
1
- = [ (7rl - y - s x y
F3
=
: ( 31
+-xT
z S(1-2).
y--
(14.33b) For antiquarks, the signs of last term are interchanged. Thus for instance, for the processes
we have for large E d2a” dx dy
+
1
GgmE 1 (1 - Y ) ~ 1 - (1 - Y ) ~
--[
2
IT
and the relation for
F3
2
2
S(1-x)
corresponding to the relation (28a) is
Deep Inelastic Scattering
500
In view of Eq. (33a) [note that, the role of ei in Eq. (28a) is taken over by trhe isospin raising and lowering operators, namely I * ] , we get FZ(2)= ~
+
FIP = 22 [u(x) d(z) F‘:
=
zF~(z)
+ c ( x )+ S ( x )+ t ( z )+ b ( x ) ]
2 [u(x) - J ( x ) + C(Z)
- S(Z)
I).(&
+ t(x) -
, (14.35a)
F,”’ = * 22 [ d ( ~+) U ( x )+ S(X) + C(Z) F3”P = 2 [ d ( z ) - u(2) s(x) - c(2)
+
+ b ( z )+ f(x)]
+ b ( 2 ) - f(2)], (14.35b)
The factor 2 is due to the fact, that, for weak dccays we have both vector and axial vector currents. The corresponding values for neutron are obtained by replacing u ++ d , H d on the ground of isospin invariance. Hence for an isosinglet, target N , we get (suppressing 2 )
[:
FIN
= 2 2 - ( ~ + i i ) +1~ ( d + ~ ) + s + b + ~ + f ]
F3VN
= 2 [;(u -
F:N
= 2
[:(u
1 2
u )+ -(d
- ‘L1)
-
d) + s + b - c - I]
+ -21( d - d) + c + t - S
-
-1
b . (14.36)
If we assume that, in a nucleon, the probability of having q and 4 ( q = s , c, b, t ) is the same or we neglect s,3 , ., then we can write a
FIN
=
E X[q+ q] E X[q q]
= F:N
4
xFIN
=
-
9
=x F ; ~ .
(14.37)
Sum Rules
501
We observe from Eq. (28d), neglecting the sea quark contxibution of heavy quarks, and Eq. (37) that, ![ = e or p]
(14.38) This ratio has been experimentally tested as the left-hand side is 1.007f0.063. This also shows that the strange quark sea contribution is very small. It verifies the charges of u and d valence quarks as their mean square is
5.
14.5
Sum Rules
One can write a number of sum rules. First, the momentum conservation gives
(14.39) where E is the fraction of the morhentum carried by the gluon constituents. Hence we get, the sum rule
4’
F,”Ndx = 1 - E .
(14.40)
Experimentally, the left-hand side is 0.52 f 0.03 giving the momentum fraction carried by the quarks. Thus the remaining momentum fraction, which is about 50%, is attributed to the gluon constituents. Since the nucleon has quantum numbers S (strangeness) = 0, C (charm) = 0, B (bottom) = 0 and T (top) = 0, we have
for q = s,c, b and T. On the other hand, the charges of proton and neutron give 1 = & q - ( u 2- Z L ) - - ( d -1d ) ] , 3 3
0 = l ’ d a [ t ( d - 2) -
(14.42)
Deep Inelastic Scattering
502
We can combine them, so that we get 1 =
A
1 =
-1
1
dx [(u- fi)
1
1
3 0
-
(d-
z)] ,
dx [ ( d - d ) + ( u - f i ) ] ,
(14.43)
thus from Eqs. (35), (41) and (43), we have
(14.44) and 1
1
F3VN(x)dz =
dx [(u - U )
+ ( d - z)] = 3.
(14.45)
If we use Eq. (35b) and the corresponding equation for the neutron, we get the sum rule (44) in the form
dx
(14.46)
X
This is known as the Adler sum rule. It is an exact slim rule obtained from quark structure of electromagnetic and weak hadronic currents and is protected by conservation laws implied by Eqs. (41) and (42). It is difficult at present, to verify it, experimentally with good precision as it requires good low x data. On the other hand, the sum rule (45), known as the Gross-Llewellyn Smith sum rule, is modified by QCD corrections in the leading order to,
s,'
F3uN ( x ) d x= 3 (1
-
q) .
(14.47)
The right-hand side of (47) for a,(Q2 % 3 GeV2) = 0.35 f 0.05 is 2.66 f0.05 while experimentally the left-hand side is 2 . 5 0 f 0.018 f 0.078, verifying the sum rule.
Sum Rules
503
Another sum rule which follows from Eqs. (28b, c) is
dx X
-
IJ' dz [U(.) + u ( z ) - d ( z ) - d ( z ) ] 3 0
1 +?l1 1 1
= 3
3
=
dz
[U(.)
- u(z)- d(z)
0
0
+ 441
d z [U(z)- d(z)]
13 + 23 J'0 dz [u(x)- d(z)],
(14.48)
on using Eq. (43). This is known as the Gottfried sum rule. Experimentally the left-hand side is 0.258 f 0.017 implying that the second term on the right-hand side is not zero. Its non vanishing does not contradict any known principle. There are two sum rules which involve the spin-dependent structure function g1(z). We note from Eq. (27) that (14.49a) where we have defined (for a nucleon target)
Here Aq is the quark contribution to the first moment of the structure function gI(x). There is also gluon contribution to it, this is due to the short-range interaction of photons with polarized gluons via the quark box diagram, shown in Fig. 5 . To include this we replace Aq by QS A@= Aq - -AG,.
2.rr
(14.50)
Deep Inelastic Scattering
504
+
Figure 5
crossed graph
The photon-gluon scattering graph.
This separation is not unambiguous but has been found useful. For the proton target Ag has been shown to be related t,o the matrix elements of the axial vector current q7,75g ( P 147F7541P) = M S J 7 4 = '1L, d , s
(14.51)
where Sp = Q 7 9 5 Q is the spin of the proton, G' being the proton spinor. For the first moment of gy(x), the gluon contribution in relation (50) is related to the triangle axial anomaly [cf. Eq. (11.79)] in the divergence of the singlet current PAO,
Sum Rules
505
Note that the second term on the right-hand side is not an SU(3) singlet. The first term on righbhand side also contains a nonsinglet part, (that is why we have piit, a subscript, q on AG in Eq. (50)) . For the proton t,arget,, Eq. (49) gives the slim rille 1
gy(z)dz = -
1 9
1 [!Ail+ 2 9
{
1 - (Ai, 12
1 9 1 - (Ail+ A d - 2As) 3
-Ad+ - A s + . . . -
Ac?) +
where - . . denotes isospin singlet sea contribution of heavy quarks and second and third terms are isospin singlets. Therefore, for the neutron target, only (Ai, - A 9 changes sign and we get in the isospin conservation limit
1’
[g?(z) - g;”(z)] dz =
1
nii - Ad
1
=-
(14.54)
since from Eq. (51), it is clear that
1
= 2 g A (sp).
(14.55)
Here gA is the axial vector coupling constant, determined from pdecay of the neutron. The sum rule (54) is known as the Bjorken sum rule. If the leading order QCD corrections are included, it then becomes
(14.56) This sum rule obtained from quark structure of electromagnetic and weak hadronic currents, is regarded as a fundamental prediction
Deep Inelastic Scattering
506
Neutron
4
El43
ry
Figure 6 Plot of versus I'y. The predictions of the Bjorken and Ellis-Jaffe sum rules are shown on the diagonal band from the lower left to the upper right of the figure. While the data and the Bjorken sum rule overlap within one sigma, the Ellis-Jaffe prediction is roughly two sigma away from the overlap region in the data, [16]
Sum Rules
507
of QCD. For g A = 1.2670 f 0.0035, a, = 0.35 f 0.05 one finds for the right hand side of Eq. (56), the value 0.187 f 0.01. The experimental situation is best summarized in the r;l,:?I plane, Fig. (6) which illustrates that Ellis-Jaffe sum rule [see below] is violated by the experimental data where as Bjorken sum rule is compatible with the data. if one One can obtain another sum rule involving only assumes exact SU(3) flavor symmetry for semi-leptonic decays of baryon octet, so that (Ail
+ A d - 2A;)
(S,)
2d5 (P IAsA P)Q2=0
=
= 9 N J = (3F - WS,)
(14.57)
where Q A , F and D have been defined in Chap. 11, namely
-
A 6 - Ad = g A = 1.2670 f 0.0035, F = 0.463 f 0.023, D = 0.803 f 0.040.
(14.58)
Thus neglecting the sea contribution of heavy quarks, one obtains from Eq. (53)
where
g:
= Ail+Ad+
Ai =A2
(14.60)
is unknown. Furthermore, if one assumes as is done in naive quark model
one has g;
73 g;
= (3F - 0 )
(14.62)
508
Deep Inelastic Scattering
so that, the sum rule (59) becomes
This is known as the Ellis-Jaffe sum rule. With g A and F / D given in Eq. (58), t,he right-hand side of Eq. (63) is 0.187 f 0.003 in disagreement with the SMC (Q2 = 10 GeV2) data which gives
ry = 0.139 f 0.01.
(14.64)
In view of the above disagreement the assumption (61) has been questioned. If one relaxes it, one does not have any prediction. together However, one can use the sum rule ( 5 3 ) , [neglecting with (64) to determine . . a ]
= 0.139 f 0.01.
(14.65)
This together with the values given in Eqs. (57) and (58) give
Aii Ad Ai
= 0.78 f 0.07 = =
-0.48 f 0.08 -0.14f0.07,
(14.66)
so that
(Aii + Ad+ AS) = 0.16 f 0.22
(14.67)
which is consistent with zero. In other words [cf. Eq. (50)] (14.68)
+ +
where AX = Au Ad As is the quark contribution to the spin of the prot,on and AG is the singlet part, of AG. Various estimates of AX indicate that AX ==: 0, which implies that (-AG) = 0.05 f 0.07. Thus one can say that, the quarks do not contribute to the spin
2
509
Deeplnelastic Scattering Involving Neutral Weak Currents
of the proton (this is known as spin crisis for the proton) implying in view of the angular momentum sum rule = AX A 6 L,, that its spin is carried by gluons and/or orbital angular momentum of its constituents. AX x 0 is in complete disagreement with NQM result which predicts AX = 1. It is important to measure both g$ and @(O) [the SU(3) singlet anomalous magnetic moment, of the proton] experimentally in order to determine the flavor and spin content, of the proton.
+
14.6
+
Deep-Inelastic Scattering Involving Neutral Weak Currents
For neutral weak currents mediated by Z-boson (see Chap. 13), the relevant Lagrangian for the processes
(v)v
+ N -+
(v)v
+x
Thus from the experimental data on deep inelastic scattering, we can determine E L ( U ) , EL(^), E R ( U ) and E R ( ~ ) . This information have been used in Chap. 13. In writing Eqs. (69), we have neglected the contribution of strange and heavy quarks. For neutron, we can obtain the structure function by replacing u ( z )* d ( z ) and u ( z ) w fqz).
We end this chapter by the remarks that the quark-parton model is simple and quite successful. A closer examination of Fig.
510
Deep Inelastic Scattering
3 reveals a systematic deviation from exact, Bjorken scaling, the structure function increases with increasing Q2 at small z whereas it has opposite behavior for large IC.The attempts to understand such deviations from the quark-parton model in terms of QCD are beyond the scope of this book.
Bibliography
511
14.7 Bibliography 1. R. P. Feynman, Photon-Hadron Interactions, Benjamin (1972). 2. P. Roy, Theory of Lepton-Hadron Processes at High Energies, Oxford University Press, Oxford (1975). 3. F. E. Close, An Introduction to Quarks and Partons, Academic Press, New York (1979). 4. G. Altarelli, Partons in Quantum Chromodynamics, Phys. Rep. 81C, l(1982). 5. D. H. Perkins, Introduction to High Energy Physics, AddisonWesley (Third Edition, 1987). 6. T. D. Lee, Particle Physics and Introduction to Field Theory, Harwood Academic (revised edition 1988). 7. T. Sloan, G. Smadja and R. Voss, The Quark Structure of the Nucleon from the CERN Muon Experiments, Phys. Rep. 162C, 45 (1988). 8. S. R. Mishra and F. Sciulli, Deep Inelastic Lepton - Nucleon Scattering, Ann. Rev. Nucl. Part. Sci. 39, 259 (1989). 9. G. Altarelli, Ann. Rev. Nucl. Part. Sci. 39, 357 (1989). 10. R. Jaffe, Lectures delivered at the ”School on High Energy Physics and Cosmology” Quaid-e-Azam University, Islamabad (March 11-25, 1990). 11. R. K. Ellis and W. J. Stirling, QCD and Collider Physics, FERMILAB-Conf.- 90/164-T (1990). 12. Small-z Behavior of Deep Inelastic Structure Function in QCD, Edited by A. Ali and J. Bartels, Nucl. Phys. B (Proc. Supplement) 18C (1990), Feb. 1991. 13. Riazuddin and Fayyazuddin, Flavor and Spin Content of the Proton, M. A. B Beg Memoral Volume (Editors A. Ali and P. Hoodbhoy), World Scientific, Singapore (1991). 14. Proc. of SLAC Summer Institute on Lepton-Hadron Scattering and Topical Conf. Aug. 5-16, (1991). 15. G. Altarelli, R.D. Ball, S.F.Orte and G. Ridolfi, “Theoretical Analysis of Polarized Structure Fhctions” CERN-TH / 98-61,
512
Deep Inelastic Scattering Talk given by G. Altarelli and G. Riodolfi at Cracow Epiphany Conference On Spin Effects in Particle Physics, Jan 9-11, 1998 Cracow, Poland. 16. M.C. Vetterli, The spin structure of the Nucleon, DESY 98211, hep-ph/9812420, December 1998.
Chapter 15 PARTICLE MIXING AND CP-VIOLATION 15.1
Introduction
We have seen that, neither parity P nor charge conjugation C is conserved in weak interaction. Let 11s consider the decay n+ --+ p+v in the rest frame of T where experimentally p+ is found to be polarized with helicity 'FI = s . p/ (PI to be negative. The application of charge conjugation operation C changes T + to T - , p+ + p- and u --+ ij but does not change the helicity. Thus if weak interaction were invariant under C, one would find Fr++p+(-)v = Fn---,p-(-)T, where (-) denotes negative helicity. Experimentally FT+-,pt(-)v >> l?n--,p-(-)T, showing that, C is violat,ed in weak interaction. If however, we now apply C P , then since helicity also changes sign we have Fn-+p-(+)F= FT++p+(--)V as seen experimentally. Thus C P is conserved here. Let, us now consider the KO -+ I?' system. In hadronic and electromagnetic interactions, the hyperchange Y is conserved so that K o ( Y = 1) t--t K o ( Y = -1) transitions are not possible. In a production process involving hadronic (or electromagnetic) interaction, KO and I?' appear as two distinctly different particles. In the presence of weak interaction, Y is no longer conserved and transitions between ' K O and Ro can occur, for example.
KO
T'T-
weak
4
weak
I?O,lAYl = 2
+
+
Thus if we write H = Ho Hw, where HO = Hhad He,,, KO and KO,which are eigenstates of H o , are no longer eigenstates of H .
513
Particle Mixing and CP-Violation
514
A linear combinates of KO and I?' will be eigenstates of H . Such states cannot be eigenstates of C or P since neither is conserved in weak interaction; C P is a better choice. Choosing the C P phase CPIKO) = - IK0) SO
that
C P IpP)= - KO)
(15.1)
7
it is easy to see that
are eigenstates of C P with eigenvalues f l . Further if served so that, [ H ,CP] = 0, then
CP is con-
so that (K! /HI Ky) = 0 = (Kf [HIK;), showing that, H i s diagonal in the basis provided by IKy) and IK;). Thus eigenstates of H can be chosen to be eigenstates of C P . Now is the antiparticle of KO ; they should have the same mass. But Ky is not, the antiparticle of K! and so they can have different properties. In fact due to weak interaction, Ky and K i should have slightly different rest energies; experimentally (mKz- mK1)/ m K and it is remarkable that such a small quantity is measured. What about their life times. Energetically kaons can decay into two or three pions. Consider 2.n final state. As seen in section 4.6, C parity of 27r state is (-l)e where I! is the relative orbital angular of 27r system. Thus N
CP
+.I)-
= (-ly(-l)*(-ly I.+z-) =
)+-.I
(-lye
=
I.'.-),
General Formalism
515
Similarly
C P IT".") = IT".") .
(15.4)
Thus only K: can decay into 27r if C P is conserved in weak interaction and K2 + 27r is forbidden. K i will have other modes, e.g. three pionic which can have C P = -1. Now decay energy available for 27r mode is about 220 MeV and for 3 pionic model it is about 90 MeV. Thus the phase space available for decay into three pions is considerably smaller than that for two pions, implying 7 1 3 T(f(10)
<< 'T(K;)
72.
Experimentally T(K:) = 0.893 x lo-'." sec. and 7 ( K i )= 0.517 x sec so that 71/72 = 1/580. As seen above, if CP is conserved, K i -+ T+T- is forbidden. But Kz + 7r+7r- occurs, showing that C P is not conserved. Numerically it is not a big effect.
A(Ki + T+T-) = 2.269 x A(Kf + T + T - )
(15.5)
This chapter is devoted to CP violation and particle mixing.
15.2
General Formalism
As seen above KO and I?." can mix. We now develop a general formalism for particle mixing. Let X o and X o be two pseudoscalar particles (X = K , B or D ;X being the antiparticle of X). Let, I@(t))be a state at time t. It is a coherent mixture of l X o ) and
F0)
+ z(t)
I @ ( t )= ) a ( t ) 1x0)
1x0)
(15.6)
where t is measured in the rest system of the particle X o . Then the time evolution of the state
Particle Mixing and CP-Violation
516
is given by
.d 9 =m 9 (15.7) dt where m is a 2 x 2 matrix in the space spanned by X o and X o states and since the particles X o and X o decay, m is not hermitian and has the form 2 ma/a = Mats - (15.8) 2 with a , a' = X o ,X o (1,2). Note that, r and 111 are hermitian 2-
If one now assumes CPT invariance, then
(X' Iml x O )
=
( X o lml X o )
or mll
= m22.
(15.10)
It is worth proving this result; CPT invariance implies (see Section 3.5).
where
and means momenta and spins of the corresponding states are reversed. Since we are in the rest frame of X o , T will reverse only N
General Formalism
517
the magnetic quantum number and so we can drop choose CP phase such that,
CPIXO)
CPIf)
m.
We may
-1XO)
=
(15.13)
= 17iPlJ).
Then
where Now Ifin)
=
Sf Ifout)
-
e2i6sIfout)
(15.15)
where 6, is the strong interaction phase for the state (14) gives finally
(foutImlXo)= r l f
e2a6a
(XoIml.Lt).
If).
Then Eq.
(15.16)
Ifmt)
In particular for single particle states Ifin) = = l X o ) [S,= 0, q;p = - 1, according to our choice of the phase in Eq. (13) so that, qf = 11, Eq. (14) gives
proving (10) and giving
Mll
= M22,
rll = r 2 2
(15.17)
that is particle - antiparticle have identical mass and same total width. Note that if we take f = X o in Eq.(14), we get, an identity
Particle Mixing and CP-Violation
518
so that, with CPT invariance alone m12 and m21 are not related. However, if we assume C P invariance, then by an argument similar to the above, the relation (16) is replaced by
(f JmlXO)= --77Ep (f Imlxo) so that, for
f = X0[q&
(15.18)
= -11
(XOImIXO)
=
(~'/rnlX'),
and thus C P invariance implies
We have the result that, in the X o - X o space m is a 2 x 2 matrix of the form
where CPT invariance alone (which we now assume) requires
A = A'
(15.21)
or All1
= M22
r l l = r22.
But hermiticity of the matrices M and I? [see Eqs. (9)] gives
M~
=
M ; ~ ,r12= rl1
M~~ = ML rll = r;, M~~ = M ; ~ rZ2 , = r;2.
(15.22)
Then the diagonalizat,ion of the matrix (20) gives the eigenvaliles Y1,2 =
A T dBC
= A =FPQ
(15.23)
General Formalism
519
where p2 = B = Mi2
i
- 5r12 i
i
c = M~~- 2
q2 =
=
- 1 7 ~ ~
q2 - Zr;,.
(15.24)
Then the corresponding eigenstates are (15.25) Hence we have the result
i M~~- Zrll- pq i iwll- -rll +pq 2
=
i 2
y1 = ml - -rl
= y2 = m2 -
i
-r2 2
(15.26)
so that ml m2
rl r2
Mll-Repq = M11+Repq = rll 2Impq = Pll - 2Impq. =
+
(15.27)
Thus finally we have
Am = m 2 - m l = 2 R e p q m1+ m2 = M11 2 A r = r2- rl = -41mpq 1 r = -(rl +r2)=rll 2 m =
Let us define
(15.28)
Particle Mixing and CP-Violation
520
If C P is conserved, then B = C , q = p ( = ~ 0) so that the mass eigenstates given in Eq. (25) become (15.30) which are now also the eigenstates of C P :
It follows that CP-violation is determined by the parameter &=-
P-9 P + i
(15.32)
Since the particles Xo and X o are unstable, it is the particles XI and X2 defined in Eq. (25) which have definite masses ml and m2 and decay widths rl and rz respectively. Let IXP(t)> be a state at time t . In the XI and X2 basis, we can write
d
idt
I*@))
=
m2 -
;r2
) I@(t)).
(15.33b)
The solution is
a(t)
=
a(0)exp
b(t)
=
b(0)exp
[ , (m2 -2
-
; PI
-r2
(15.34)
Suppose we start, with Xo,viz. IQ(0)) = lXo), then from Eq. (25), we get,
a(0) = b(0) =
JIPI'
+ Iq12 2P
(15.35)
General Formalism
521
Hence from Eqs. (33a), (34) and (35), we get,
IV>>
+ exp [ (--im2- -rz 2 t /x2) l ) 1 IW))
{
= 2 exp(-iml-
-2P [exp (-iml
1
(15.36a)
1
1
-rip 2 + exp (-im2 - -r2) 2 t ) 1x0) -
-rl) 1 t - exp (-im2- -r2) 1 t] 1x0). 2 2 (15.3613)
Equation (36b) clearly shows the particle mixing. Similarly if we start with X o we get, at time t:
(W)) - exp
[ (-im2 -
(15.37a)
IW)
-{1 P- [exp (-iml- -rl 2
(
2 '
- [exp (-iml- -rl t + exp -im2-
(
2
) ] Ix -r2 ) t] I x-"} . 2
t - exp -im2 - -r2 t 2
O)
(15.3713)
xo
From Eqs. (36) and (37) we can determine X o and mixing. It is clear that if we start with X o , then at time t , the probability of finding the particles Xoor is given by [using Eq. ( 3 6 ~ 1 = e-rzt 2e-"t cos Amt] 4
xo
I(xoI+(~))/~
+
+
Particle Mixing and CP-Violation
522
(15.38) We define the mixing parameter r as (15.39) where T is a sufficiently long time. In the limit, T (38)' we can easily determine:
r=
2+y2 Il+&l 2 + x 2 - y 2 1 - E 2 -
--f
m,using Eq.
(15.40)
'
where x = A m / r and y = Al?/2I'. If we start, with X o , we can use Eq. (37b). Then we find
When CP-violation effects are neglected, then (15.42) The asymmetry paramet,er a (15.43) is a measure of CP-violation. We define another parameter x which is also a measure of particle mixing. Let x be the probability of X o --+ then
xo,
General Formalism
523
Thus
(15.45) Similarly, we get
We note from the definitions 2 =
9,y =
217
0 5 22
(15.46)
0bviously OIr51 (15.47) We now discuss, how the mixing parameter r can be measured experimentally. Suppose that X o and X o are produced in the reaction e-e+ -+ X O
Xo.
Taking into account the particle mixing, we have four possible final states X o X o , X o X o , X o X o , X o X o . Experimentally X o X o and x o X o are indistinguishable. We can define a parameter
(15.48) which can be measured experimentally. N ( X o X o )can be identified If by some convenient final states (e.g. two charged leptons L-L-). X X o pair is produced incoherently (for example not through a resonance of definite spin and parity and C-parit,y), then
(15.49) Neglecting CP-violation effects, i.e. using
= I
2r
+ r2'
x = x,we get (15.50)
Particle Mixing and CP-Violation
524
Now suppose that X o X o are produced through a resonance with J p c = 1--, for example
e-e+
--t
T t BOB'.
For this case we have to consider a state with C
= -1 viz.
[ I W Ip(t)) - (pw)IW)] *
If the two decays take place at
and t 2 , then neglecting CPviolation, we have from Eqs. (36) and (37): tl
where
k exp( S A m t ) exp 2
Hence we have
(15.52)
CP-Violation in the Standard Model
1
+ exp (-:Ar(t2 = 4[ r2
2
525
- tl) f 2 Reexp (-iArn(t2 - h))
*
(15.53)
- $(ar)2 r2+ ( A v z ) ~
(Xoxo) and N ( X o x o )=
Noting from Eq. (51) that N ( X o X o )= N N ( x o X o ) ,we get [cf. Eq. (40) with E = 01
R=
(Am)2+ :(Ar)' = r. 0 2r2 ) + ( nm)2- (nry
N ( X o X o )-
~ ( ~ 0 x
;
(15.54)
15.3 CP-Violation in the Standard Model Here we discuss how CP-violation can arise in the standard model of electroweak [SMEW] interaction. Three generations are known to exist:
Since the mass eigenstates are not identical with weak eigenstates, the hadronic charged current, can be written as [see Sec. 13.101.
J,W(H) =
c Kqk
7p
qL
(15.55)
i=u,c,t
where V is the CKM matrix. This gives the charged current, interaction Lagrangian
(15.56) Now [see Sec. A.81
Particle Mixing and CP-Violation
526
Noting that q the phase q(W) q ( q ) q * ( i )can be chosen to be +1, Eq. (56) gives
CPL,, ( C P )-
This is identical wit,h (56) except that,
v,,
-+
K:,.
(15.59)
On the other hand
where L,, is the neutral current interaction Lagrangian, involving only diagonal couplings,
(15.61) Thus t,he neutral current, in the interaction Lagrangian is necessarily C P invariant,. On the other hand from Eqs. (56) and (57), it is clear that, ( c P ) L c c ( c P ) -= l L,, (15.62) if and only if V is real [K, = V,:] or can be made real. Thus SMEW is capable of CP-violation. Suppose we have N generations so that, V is an N x N matrix and as such has N2complex elements or 2N2 real parameters. But since V has to be unitary, it has N 2 real parameters. Then there is freedom to define any quark field by a phase, for example, d -+ eied, WpKdiL~pd4 L W p & i L T p ( e " e d L ) = Wp(l&eie)(iLypdL) and eie can be absorbed in the redefinition of V , d without, changing the
BOBo Mixing and CP-Violation
527
physics. Thus phase of any individual CKM matrix has no physical meaning [what counts is the relative phase]. Hence the number of phases which have no physical meaning [remember there are 2N fields] are (2N - 1). Therefore, number of independent, parameters N in V N ~are
N 2 - ( 2 N - 1) = ( N - 1)2 0 N = l
1 N=2 4 N=3
(15.63)
One way of choosing t,he parameters is mixing angles and complex N orthogonal matrix, then the number of phases. Now if V N ~were independent parameters would be N 2- N = which give the number of mixing angles. Then the number of phases are
v,
( N - 1)2 -
N ( N - 1) 2 =
-
( N - 1)(N- 2 ) 2
0 N=l 0 N=2 1 N=3
Thus the SMEW interaction is capable of C P violation provided that V is at least 3 x 3 i.e. three mixing angles and one phase. In other words C P violation can be accommodated if number of generations is at least, three. This observation was made before the third generation was discovered.
15.4
BOBo Mixing and CP-Violation
We now apply the general formalism to BOB' system. We can write 2
~ 4 -, -rI2 ~ = (EOl~,aff~=~pO). 2
(15.64)
Particle Mixing and CP-Violation
528
Figure 1 Box diagrams for
lABl
= 2 transition.
The H,affEz2induces particle - antiparticle mixing involving the neuBf. We shall deal with both tral B mesons, B: -+ B: and Bf of these transitions. The H$BB'2 for B: --+ B: transition can be extracted from the box diagrams of Fig.1 [for @ + B:, change d + s] as in the standard model AB = 2 transitions arise from these diagrams. We note from --f
that only Bo and Bo decays contribute to FI2. The common final states for the I?: and B: decays are shown in Fig.2 while those for B: and B: can be obtained by changing d to s. We will not, write H,affB=' explicitly. Some of the main conclusions can be deduced from the general features of the diagrams:
BOBo Mixing and CP-Violation
529
C
B0 .-$Ti d
Figure 2
C
Diagrams showing the common final states for the BZ and
B: decays.
Particle Mixing and CP-Violation
530
i) From Figs. 1 and 2, we note that, M I 2 and I’12 depend on vqb ( q = t , c , u ) . These parameters are given by the matrix elemenh of Cabibbo-Kobayashi-Makawa (CKM) matrix parameterized in the Maiani-Wolfenstein way: v q b
vs
v$ or
M
(
1 - ;A2 -A
X 1 - 5.A 1 2
AX3(/, - iq)
AX^
1
~ ~ 3 ( 1 - iq)
AX2
) (15.66)
where X z sinB, z 0.22 and 1Al = 0.90 f 0.18 is determined from semi1ept)onicB-decays; q # 0 if C P is not conserved. The unitarit,y of V gives
vu:
vub
+ v,*,
vcd
+ y>&b
=
0.
(15.67a)
To leading order in A, tthis relation can be written, using Eq. (66), as v,*, &d - Kb = 0. (15.67b) The relation (67) can be represented by a triangle in the complex plane (Fig.3). In particular we note that
+
2rl (1 - PI (15.68) q2 (1 - p)2 . ii) If the quarks, t , c, u have nearly the same mass, sum of their contributions would involve ( 1 vqd Vgtb)’ which vanishes sin
2p
=
+
q=u,c,t
due to the iinitarity condition (67). Sincc rnt >> mC,it is clear that, dominant contribution comes from exchanged t -quark. iii) In the standard model all the complex phases enter through CKM matrix (see Eq. (66)). In particular arg [A(&
-+
Bd)]
= arg = arg
I$[
[ W d
y;,2]
BOBo Mixing and CP-Violation
531
Figure 3 The CKM-unitarity triangle in the Wolfenstein parameterization.
(15.69a) where
~FKM = arg(&:&b)
= p.
(15.69b)
&om the above considerations, using Eqs. (66) and (67), one finds the main results from the box diagrams:
BjBj System: M12 0: (vtbV,*,)2
mi = [AX3 (1 - p
+ iq)I2 m:
(15.70a)
while [cf. Eqs. (65) and (67a)l r12
0: ( K b
=
v,*, + vub vtd)2mi
(I& V,+,)2V L ~= [AX3 (1 - p
+ iq)J2mi.(15.70b)
Hence 1M121 >> IF121 and both A412 and I'12 have the same phase in the leading order. Thus it follows from Eq. (69):
M12 = lM12(e2ip,r12=
lr12le2@,
(15.71a)
Particle Mixing and CP-Violation
532
and
(15.71b)
B,OB,O Systems:
Hence we have
(15.73)
We also note that
(15.74) Using Eqs. (24) and (71), we get for the Bj that A 4 1 2 and l712 have the same phase).
- Bj system (noting
(15.75) From Eqs. (28) and (75),we then obtain
(15.76) Hence [cf. Eq. (73)l
Ar Am,
<< 1.
(15.77)
BOBo Mixing and CP-Violation
Now IAmBI N 3.7 x we conclude that Am,
N
533
MeV, and and
r
-
5.9 x 10-l' MeV, hence
Ar
r << 1.
(15.78)
For the B: - Bf system we see from Eqs. (24), (28), (72) and (73) that Eq. (77) also holds for this system. There is general consensus that the same is also true for the inequality (78). So the mixing effects in these cases can be interpreted in terms of AmB/r. From Eqs. (29) and (71), we have for the Bj - B j system 4 - e--22P P
,
(15.79)
and hence Re
E = 0,
Im
E
= tan@.
(15.80b)
Since sin p is expected to be small, we have Ree = 0, Im
E M
sin
p.
(15.80~)
Similarly for the Bf - I?: system q / p M 1 and E M 0 [cf. Eqs. (29) and (73)]. Thus E is expected to be small in the B syst,em. In fact, the theoretical estimates give E
0(10-3) for Bd = 0(10-~) for B,.
=
(15.81)
Thus it seems that the prospects for observation of A B = 2, C P violating phenomena are essentially hopeless. However, in the B-system, CP-violation in B decays ( A B = 1) can be large, unlike in the kaon system [see Sec. 51. We now discuss the CP-violation and the mixing in Bo decay. First we consider the case in which the final states f and
Particle Mixing and CP-Violation
534
# f.
f are
not eigenstates of C P , f amplitudes
Let us define the decay
and the ratios
Then from the decay B o ( t ) f and its CP mirror Bo ( t ) + j , w e have from Eqs. (36b) and (37b), using Eqs. (28) and (78) --$
-
Am
e
cos -tA(f)
2
Am - i - sin -tA(f)[ P 2 9
(15.84a) -
rf(t) =
1
2
ePrt cos T t
Am
A(f) - z-sin-tA(f)l P Am - -
On using Eqs. (82) and (83), we obtain
4
2
. (15.8413)
BOBo Mixing and CP-Violation
535
Now from Eq. (16) with X = B , m = H w , CPT invariance gives
Similarly = rlf
e2&
A*(f).
(15.86b)
Thus from Eq. (83)
Hence we can write
where $f is the weak phase of the decay transition. Then Eqs. ( 8 5 ) , on using Eqs. (79), (86) and (88), give
I'f(t)
c(
1
+ lzfI2)+ (1- I z ~ ~ ~ ) c o s -21zfl sin Amtsin(24 + q5s)} IA(f)I2
e-rt {(I
-21Zfl sin Amtsin(-24
+ 4,))
where 2 4 = 20 + 2 4 f ,
(15.89) -
4 5
=
Amt
6,- 6,.
(15.90)
We have introduced four parameters IA(f)l, Izfl, 4 and &, Can these parameters be determined from the decay rates (89). The
Particle Mixing and CP-Violation
536
d
Figure 4
a
Quark level diagrams for B
-+
p7r decays.
answer is positive since the decay rates are not, numbers but, are funct,ions of time. As an illustration of the above considerations, consider the decays &(t) t p+n-(f) and &(t) --+ p-n+ (f).The quark level diagrams for these decays are shown in Fig. 4. Thus
so that 2$f =
+ 6AKMand using Eq. (69b), we obtain
BoEo Mixing and CP-Violation
537
so that
[-
= arg
Vud
-v1zb
(15.93)
%d
or using Eq. (66) and Fig. 3
4 cx arg
(z)
= a.
(15.94)
We now apply the above formalisms when the final state in Bo decay is an eigenstate of C P
If) = c p If>= d
P
If>, 77iP = f l .
If)
(15.95)
Then it follows from Eq. (83) that z ~=f l / Z f , so that Eq. (88) gives I ~ f =l 1, 43 E 6,- 8, = 0, or r, (15.96) accordingly as Eqs. (89)
T],&~ =
f 1 [cf. Eqs. (86)]. Then it follows from
Then the asymmetry is given by
Particle Mixing and CP-Violation
538
As qAP (31)is known and Am is already measured, this asymmetry measiires 4, independent of strong phases. The time integrated CP asymmetry is (15.99a) which on using Eqs. (97) gives
Af
= -qLP
I? sin 24
Am F2 Am2 X
f sin24 = -vcp
+
(15.99b)
___ 1f X 2 '
For B0 (BO) II,Ks,T A P = l l C P ($) YlCP ( K , ) = (+l)(+l) = 1 and in the standard model from the transition 6 CCS, it is easy to see that, @f = 0 since the CKM elements involved are Vz V,, which according to Eq. (66) do not, involve any CKM phase. Thus according t,o Eq. (90) +
--f
4 = P.
(15.100)
Thus from Eq. (98) the asymmetry is given directly in terms of CKM phase /3 (independent of strong phase).
A Q K (~t ) = - sin 2P sin Amt
(15.101)
and the time integrated asymmetry is A + K s = - sin20
If p
X ~
1 +x2'
( 15.102)
2! 10' and x d N 0.7 (see below), then the asymmetry IA,,+K~I N 0.16, which is much larger t8hanE N in kaon decays (see Sec. 5).
BOBo Mixing and CP-Violation
539
Finally we consider the decays [for example X- = D-, X+ =
D+l
In the standard model Bo decay into P v X - and Bo decay into e-a X + are forbidden. Then A ( f ) = 0 = A(f) i.e. zf = 0 = xf so that from Eq. (85). 1
r ( B o ( t )+ f )
+ cos Amt]
cx
IA(f)12e-rt[I
=
r (B0(t)+ f ) .
On the other hand from Eq. (84a) [replacing f by
( B V ) -+ f)
(15.103)
f]
.q Am IA(f)lZe-rtI - 2- sin - tI2 P 2
CX
1 2
-IA(f)I2 e-rt [I - cos Amt] (15.104) where we have used
(A(f) 1
= ( A ( f ) ( Integrating . Eqs. (103) and
(104)
n
=
-
2
X '
+22 =r
[cf. Eq. (40)].
(15.105)
Hence a nonzero value of S would indicate Bo and Bo mixing as in the standard model z~ = 0 = Zf = 0. But if xf and Ef are not zero due to some exotic mechanism, then 6 # 0, even if there is no mixing.
Particle Mixing and CP-Violation
540
The particle data group give for the Bj
-
B: system the
value Xd
=
=
rd
1 rd
+
F(P+
x-)+
x-)
r(p+ ryp0.172 f 0.010
x+) (15.106a)
and xd = 0.723 f 0.032.
(15.106b)
This gives a clear proof of the B: mixing in the standard model. However, in contrast, to KO - K'case (see t,he next section) 6 does not give any information about CP-violation. The value xd provides a determination of I& in terms of mt and fs 6( G J B ) which paramet,erizes the hadronic matrix ~p bLa) ( 2 elements of the four quark operator [cf. Fig. 11 yp bLc) between Do and Bo. This can be seen as follows: First we note that the dominant contribution to MI2 comes from the top quark in Fig.1 and it has been shown that [cf. Eqs. (76), (77) and (10511
(15.107a) where 1 F(x) = -
4
+ 4 ( 19 - ~ )- _32 (
1 3 x2hx - -~ l - ~ ) ~2 ( 1 - ~ )
(15.107b)
where ~8 is a QCD correction factor. The constant
*
~~
BOBo Mixing and CP-Violation
541
The measured value of xd thus provides a determination of terms of mt and c B d , yielding
&d
in
which lies within the standard model iinitarity constraint, 0.013 [cf. Particle Data Group].
&d
<
For Bf
- @ mixing parameter (15.109a)
we have [using the analog of Eq. (107a) for x s ]
( 15.109b) First we note that since xs/xd 1/x2 = 1/sin28, M 21, x , is expected to be large. The ratio (109b) is a useful quantity since it leaves the square of the ratio of CKM matrix elements, multiplied by a factor which reflects flavor SU(3) breaking effects which we lump into a parameter (f. The particle data group gives the lower limit for x , , x s > 14, which together with x d given in Eq. (106b) yields x,/xd > 19.4. This bound has been used to restrict the allowed p - q region for some representative values of [S. This results in the range N
0.2 < 7 < 0.4 0 < p < 0.4 with the best solution around p = 0.11, 7 = 0.33.
(15.110)
Particle Mixing and CP--Violation
542
Coming back to the ratio ( x c s / x din ) Eq. (109b), we see that, most of the models give ( J B , / J B ~ )= ~ 1.2 - 2. Thus taking T B ~M T B ~VZB, , Mm ~( J B~, / J ,B ~ )M 1 and constraint (110), we can safely conclude that, x, >> 2 so that, Eq. (105) gives r, = 1 and hence from Eq. (45), x, = 0.5. Any marked deviation from x, = 0.5 would indicate some new physics beyond trhe standard model. The particle data group gives xs > 0.4975, consistent, with the above standard model value. We conclude this section with the remarks that there is clear experimental evidence for Bo - Do mixing. The experimental determination of asymmetry parameter AqKS[cf. Ey. (102)] can give us information about the angle p. 15.5
CP-Violation in K°Ko System
We now apply the general formalism developed in Sec. 15.2 to the K0Ko system. Here we denote K1 and Kz as K s and KL. First we discuss hypercharge oscillations. Suppose that, at, t = O,Ko (Y = 1) is produced by the reaction 7r-p --t KoAo.The initial state is then piire Y = 1. It is clear from Eq. (36b) [with X = K ] that, a kaon beam which has been produced in a piire Y = 1 state has changed into one containing bot,h parts with Y = 1 and Y = -1. Experimentally gocan be verified through the observat,ion of hadronic signat,ure such as I?" p + 7r+ A' since 7rf Ao can only be produced by '?I and not, by KO. The probability of finding Y = -1 component, at time t in the kaon produced at, t = 0 in a pure Y = 1 state is given by Eq. (38) [I&] << 11.
P(K0
-+
KO, t )
If kaons were stable
N
1 4
oo),then 1 P ( K o -+ KO,t ) = -[I - cos Amt] 2 (TS,TL
-+
CP-Violation in K°Ko System
543
which shows that a state produced as pure Y = 1 state at t = 0 continuously oscillates between Y = 1 and Y = -1 states with circular frequency w = Am/h. Kaons, however, decay and oscillations are damped. By measuring the period of oscillation given by 27r/(Am/h), one can determine Am. system. From We now discuss CP-violation in KO and CKM matrix [cf. Eq. 661, we have
vts~; =
+
- ~ ~ ~ 5 ( 1 - iq)
[-A
vusv;*
=
x (1 -
+ A2A5(1 - p + iq)] (15.112)
f) 1
+
+
where we have used l(-d = -A A2X5 (1- p i q ) as given by the unitarity of CKM matrix. It is clear that. top quark contribution to ReM12 as compared with that of the c quark is of the order of XlOma M 5x and hence is negligible for mt x 175 GeV. mc Thus since M12 and r12 cx (vCsV$)2we conclude from Eq. (112):
3
+
Im rI2 << Re r12 Im M12 << ReM12.
(15.113)
Now from Eq. (24), we have
i Im M12+ Im r12 Re MI2 - $ Re rI2 1/2 (15.114a)
1
so that
+
p 2 - q2 iImM12 1 Im r12 2E - - _-1 &2 p2 q 2 ReM12 - $ Rer12 '
+
+
(15.114b)
Thus we get from Eq. (28), using the approximations (113) m~ - m s = 2Repq = 2ReM12 A r = r L - r S = -41mpq = 2 R e rI2. (15.115)
Am
E
Particle Mixing and CP-Violation
544
Hence we get, iImM12 &=
+
Im
r12
(15.116)
Am-gAI'
The parameter E determines CP-violation in KO - Ei' mixing. However, C P violation can also occur in the decay amplitudes KO -+ 27r and Ei'O -+ 27r. Now two pions in the final state can either be in I = 0 or I = 2 state. The dominant decay amplitude is for I = 0 due to A I = f rule, )A2/AoI N $. We define the decay amplitudes: (27rOut,I
= OIHIKO) = A0
(27rOut,I
= 21HIK0) = A2 ei62.
ei60
( 15.117)
Here 60 and 62 are the phase shifts for I = 0 and I = 2 7r7r scattering. We take the S-matrix for these states S = e2i60 and e2i62
Now CPT invariance viz. Eq. (86a) [with Bo replaced by KO and phase qf chosen to be -11 gives
( 15.118) Using the Clebsch-Gordon (CG) coefficients, we have
A(Ko
-+
T'T-)
=
1
fi
1
A ( K o -+
n + ~ -= ) --
A(Ko
-+
7r 7r
A(Ko
-+
x07r0)=
fi
0 0
[;."
-et60
1
e i60
) = - ei60
4
-
1
[A0 -
- ei60
fi
+ FA21 [&A;, + FA;] A0
[A:
& FA21 -
&FA;] ,(15.119)
CP-Violation in
KOK'
System
545
where F = ei(62-60). Ignoring the irrelevant, overall phase factor ei60, we have from Eqs.(25), (29) and (119)
e,
and E E where we have neglected corrections of order E fi (in fact the last two are completely negligible, much smaller Re A0 than the first one) and we have defined
-
&
=
&'
=
&+i-
Im A0
(15.122a)
Re A0 Re A.
Re A. (15.122b)
The quantities Im Ao, ImA2, and Im E depend on the choice of phase convention. It, is possible by a choice of phase convention to set ImAo = 0, known as Wu and Yang phase convention, in which case F = E . Note, however, the value of E' is independent of phase convention. Its nonzero value would demonstrate direct CP-violation. We now adopt Wu-Yang convention so that E=E
(15.123a)
and (15.12313)
Particle Mixing and CP-Violation
546
If we retain only the dominant two-pion contribution to the unitarity relation (65) for r12, we get on using Eq. (119)
(15.124) This gives (15.125) where rll = r + r 5801. Hence
pv
rS/2N
-?,since rs >> rL (rs/rL
N
which is completely negligible. Hence we get, from Eq. (116) tan
2Am
QE M --
nr
(15.126)
which is clear from its derivation is a consequence of CPT invariance and the unitarity relation (65) approximated by dominant two-pion contribution. Putting the experimental values
Am = (mL - m s ) = 0.474 rs nr = rL-rs=-o.998rs we obtain the phase of
(15.127)
E
& = 43.49 f 0.08'
(15.128)
while Eq. (123b) gives 4Ej
= S;! - So
+ -7r2
N
48 f'4
( 15.129)
CP-Violation in KOKO System
547
where the numerical value is based on an analysis of Finally writing Eqs. (120) and (,121) as
7r 7-r
scattering.
the experimental measurements give:
1q+-1 $+-
(2.285f 0.019) x = (43.5 f 0.6)’ lqool = (2.275 f 0.019) x boo = (43.5 f 1.0)O A$ = $o, - $+- = (-0.1 f 0.8)’. =
Since E‘ involves the AI = [cf. Eq. (123b)l one has
x
(15.131) (15.132) (15.133)
f rule suppression factor IA2/AoI
IE’I << (€1.
M E 1
.
Then from Eq. (130)
1 - 6Re(:)
(15.134)
The measurement of R provides a test for [ A S [= 1, CP violation. Recall here that q / p = (1 - E ) / (1 E ) , K L K2 E K1 [cf. Eqs. (2) and (25)]. Thus if E’ = 0, CP-violation arises entirely through the mass matrix i.e. through lASl = 2 transitions KO t--) I?’ allowed by second order weak process. This is accommodated in superweak theory. In case Re(&’/€) is not zero, CP violation must occur in a decay amplitude [specifically viz. Im A2 # 0 cf. Eq. (123b)l as well as through the mass matrix. The present experimental value is for E’IE
+
E’
- = (1.5 f 0.8) x &
N
+
(15.135)
Particle Mixing and CP-Violation
548
Finally the phases $+- and $oo can be used to test C P T symmetry. From Eqs. (130), we find
A$
N
3 Re
(3 -
tan($,
-
&).
(15.136)
Using Eqs. (128), (129) and (135)) one can limit, the magnitude of the right hand side (which has negative sign) to be under 0.06’ showing the experimental value for the phase A$ in Eq. (133) to be consistent with CPT, although further accuracy will be desired. Finally what are theoretical expectations for the ratio &’I&. First we note that in the standard model, the tree level diagrams d : V,, and A; so that shown in Fig.5 involve CKM elements A, = v A(KL -+ T+T-)involves (A, - A:) = Im A, = 0 [cf. Eq. (66)]. Thus (&’I&) arises from the ratio of the so-called “penguin” diagram shown in Fig. 6 to the box diagram shown in Fig.1 with b replaced by s. The CP-violation is determined by Im X i = Im At, [cf. Eq. (66)l where X i = K d V,:, z = u , c , t . This involves t quark which belongs to the third generation and which has very small mixing with the first and the second generation. This also explains why CP-violation is so small in kaon decays. The theoretical prediction for Re (&’I&) is not precise [for mt > mw] but most, theoretical calculations give
xi
Re
():
<3x
(15.137)
since this ratio depends on various parameters which are not as yet well determined. This is consistent with its experimental value given in Eq. (135). Finally we discuss the CP asymmetry in leptonic decays of kaon. Let us define the decay amplitudes: (1 = e,p) (15.138)
CP-Violation in
KORO System
Figure 5
Tree level diagrams for K --+ 27r decays.
549
Particle Mixing and CP-Violation
550
W 71-
Figure 6
Penguin diagrams for K -+ 27r decays.
CP-Violation in K o R o System
551
Then CPT invariance gives
KO
--t
KO
+ 7r+
7r+
+
+
+ 77: f * + n:g*.
11-
( 15.139)
Hence from Eq. (25), we get,
A(K; + K
+ i+
+v)
=
A(K;
+ t- + 77)
=
+T+
P f +49
JWi2 ”* +‘.f* . Jrn
(15.140)
Thus the CP--asymmetry parameter 6,can be written as
(15.141) Now using Eq. (29), we get 61 M
2Re~ 1
1 - 1X1l2
+ 1x112+ 2 I m ~ I m x ~
(15.142)
where we have put 9
7’
x1 = In the standard model, xl = 0. Hence we get
6, = 2Re
E .
(15.143)
(15.144)
The experimental average for this quantity is 6l = (3.27 f 0.12) x
(15.145)
Particle Mixing and CP-Violation
552
Using I E ~ M lq+-/ and the experimental values of q+- amd & given respectively in Eqs. (131) and (128), we get 2Re E = (3.32 f 0.03) x lop3 = 61 in excellent, agreement with Eq. (145). We conclude this section by noting that, from Eq. (40), the mixirig pararnct,er T for thc kaori: (15.146) since for kaon
(15.147) Thus mixing parameter T K has the maximum value viz. unity to be compared with Td = 0.20 & 0.01 [cf. Eq. (106)l.
15.6 CP-Violation in Hyperon Non-Leptonic Decays Because of the uncertainities in E’, it, docs not, as yct, test, direct CP-violation in the standard model. There is t h i s a need to study CP violation oiit,side the kaon system, we have already studied CP violation in B decays. We now consider the same in hyperon decays: A 4 px- and E- 3 AT-. These are described by the amplitude [cf. Sec. 11.2.31 a, +up o n, n = p’//p’J,p’ being the momentum of the final baryon. The observables are decay rate I?, asymmetry parameter a , the tranverse polarization of final baryon 0 and the longitudinal polarization of final baryon y defined in Sec. 11.2.3. Let us now construct, CP-odd observables i.e. compare B BITwith B --+ BIT+. C P symmetry predicts --f
-
r
= r , E = - ff,
p
=-
p.
(15.148)
Thus to leading order, CP-odd observables are
(15.149)
553
CP-Violation in Hyperon Non-Leptonic Decays
Such asymmetries can be measured in the proposed Super Lear in
where one studies the asymmetry
Here N,' is the number of protons with (pi XPA) pf greater than or less than zero. FA denotes the polarization of A. Similarly in the reaction
(15.152)
the relevant asymmetry is
=
-P~aA&(Sa,+ S&) IT
8
(15.153)
where Np+ denotes number of events with PE (pf x p ~ greater ) than or less than zero. We now discuss the isospin analysis. First, we note that, assuming CPT only, Eq. (86) gives
where be (I) are strong phases. Selecting the phase qf as (-l)'+', Eq. (154) gives Ze(1) = (-1)
t+ieaiat(r) -*
ae (1).
(15.155)
554
Particle Mixing and CP-Violation
(15.156)
( 15.157) To leading order, we obtain
since there is only one isospin final state in 2 decay and neglecting and which are very small ( w &) if A1 = f rule dominates, we get,
z,
6a
Sp
(6: - 6;) sin (8 - $:) cot (6: - s;)sin( 8-4;).
= -tan
=
sr A + ~ ~ 10-6 Z - + AT-
0
sa
sp
10-4 10-4
3 x 10-3 10-3
Problems
555
These estimates have considerable uncertainty and are model dependent. The above observables can also be used to study the effects of extensions of the standard model. 15.7
Problems
1) Show that if CP is conserved, then
K;
-+7r+7r-7ro
is allowed for pions in I = 1 , 3 states with C = L
K:
=
even and
tn+7r-7ro
is allowed for pions in I = 0,2 states with 'k = L = odd, where L is the relative orbital angular momentum of n+7r- system and 'i is that of xo relative to the center of mass of 7r+ and 7r-, Show that
K:
+ 7r07r07ro
K:
+7r7r7r
is forbidden but 0 0 0
is allowed. 2) F'rom the unitary triangle, show that
sin2a = sin2P
=
sin27 =
3) Consider the decays B* + ?yo K'. Draw the tree level and penguin diagrams for these decays. Denoting the contributions from these diagrams respectively by
~(f),&,,
ei6a
, A'(f)eZ6gK,v cia:
556
Particle Mixing and CP-Violation
where f = # K + and ~ C K Mis the phase of some product, of CKM elements and 6, are strong phases. Using CPT invariance, write the corresponding contributions for B - + f, f f .rr0K-. Show that, CP violating asymmetry,
(,, f - 6gKM) and 4, = (6,- 6;) . Note that if $3 is where q!I = & absent, CP is not violated; in fact, both CKM and strong phases are needed.
Bibliography
557
15.8 Bibliography 1. CP-Violation (Editor: C. Jarlskog), World Scientific (1989); C P Violation in Particle Physics and Astrophysics, edited by J. 'Ran Thanh Van, References to earlier literature on C P Violation and to original papers can be found in these books. 2. 1.1. Bigi, V.A. Khoze, N.G. Uraltsev and A.I. Sanda, ref. 1. 3. K. Kleinknecht, ref.1. 4. G. Altarelli, Lectures at Cargese 1987 School on Particle Physics (CERN-TH-4896/87). 5. J.L. Rosner; Heavy Quarks, Quark Mixing and CP Violations, in Testing the Standard Model [TASI 901, Editors M. Cvetic and P. Langacker, World Scientific, Singapore, 1991. 6. Y . Nir, The CKM Matrix and CP Violation, in Perspectives in the Standards Model [TASI 911, Editors R.K. Ellis, C.T. Hill and J.D. Lykken, World Scientific, Singapore, 1992. 7. B. Winstein and L. Wolfenstein, Rev. Mod. Phys. 6 5 , 1113 (1993). 8. A. Ali, in B-decays (Revised 2nd Edition) editor: S. Stone, World Scientific, Singapore, 1994. 9. Y. Nir and H.R. Qusi, in B-Decays, (Revised 2nd Edition), editor S. Stone, World Scientific, Singapore, 1994. 10. S. Pakvasa, CP Non-conservation: The Standard Model; S. Okubo, CP-Violation in D* and B* Boson Decays, in A Gift of Prophecy, Editor E.C.G. Sudarshan, World Scientific, Singapore, 1994. 11. S.V.Somalwar, CP/CPT Experiments with Neutral Kaons or Experimental Study of Two Complex Numbers q+- and 7 0 0 ; G. Valencia, Constructing CP-odd observables, in C P Violation and the Limits of the Standard Model [TASI 94), Editor J.F. Donoghue, World Scientific, Singapore, 1995. 12. B. Kayser, "CP Violation and Beauty" Lectures presented at Summer School in High Energy Physics and Cosmology, International Centre for Theoretical Physics, Trieste, 12 June - 28 July 1995 [SMR.856-341.
558
Particle Mixing and CP-Violation 13. J. Buras, R. Fleisher, hep-ph/9704376, in Heavy Flavours 11, A.J. Buras, M. Lindner (Eds.), World Scientific, 1997. 14. Particle Data Group, C Caso et al; The European Physical Journal C.3 (1998) 1. 15. M. Witherell, The B-Physics Overview, SLAC Summer Institute on Lepton-Hadron Scattering Aug. 5-16, 1991, Stanford, CA.
Chapter 16 WEAK DECAYS OF HEAVY FLAVORS In the standard model, three generations of matter replicate themselves with increasing mass scale. We have already discussed the first and second generation leptons ( e , ve), ( p ) vp). In this chapter, we first discuss the weak decays of T lepton (the third generation lepton). Later we study the heavy flavors viz. decays of D and B mesons. The study of heavy flavors provides us an opportunity to discover any deviation from the standard model. However, we will find that the standard model works quite well for heavy flavors. +0.30 We begin with r-decays. The mass of T lepton is 1777.00 -0.27 s. The upper limit MeV and its mean life is (291.0 f 1.5) x on the mass of v, is 24 MeV.
16.1 Leptonic Decays of
T
Lepton
In the standard model, the third generation leptons ( 7 , vT) behave exactly in the same manner as ( p , vp). Because T lepton mass is and K’s). As far as 1777 MeV, r can decay into light mesons (A the decay Tu, e- V e is concerned, in the standard model it should have exactly the same structure as that for the decay 1-1- + u,, eDe. Now e- and ve are common in both these decays. The e- and Ve enter in the effective Lagrangian for muon decay in the form E-yp(1 - y5)ve, it should occur in this form in the effective Lagrangian for r-decay. Hence the most general form for
-
+ +
+ +
559
Weak Decays of Heavy Flavors
560
the T-matrix is given by:
where p l , p2, k l , and k2, are four momenta of 7 , v,, e and V , respectively. In the standard model, E = +l. Thus any deviation from the standard model should manifests itself with a value of E different from 1. Using the standard techniques, we can easily calculate the electron energy spectrum
+
dr z-G’m’x2 [(l ~ ) ~-(2x) 3 dx 3 8 4 ~ ~
+ 6(1 - ~
) ~- (z)1].
(16.2)
In deriving the above expression, we have taken neut,rinos to be massless and have piit, m,/m, z 0 and II: = 2E,/m,. It, is convenient to put Eq. (2) in the form (16.3) where
rn: rz-G$ 1 9 2 and
+
(1 c2) ~ ~2
+
3 (1 E)2 1+&2
p=s
(16.4)
(16.5) ‘
Equation (4)gives the decay rate for the decay 7 - -+ e- + u, + D,. Equation (5) gives the Michel parameter p. In the standard model [ V - A theory ] E = +1, p = The experimental value for p is 0.742 f0.027 in agreement with the t,heoretical value of 0.75. This reinforces our assumption that (7,v,) are sequential leptons.
i.
Leptonic Decays of
Using
Since (1 ratio
B R (T
T
E =
1, we get from Eq. (4)
+ 6;ad)/(l + 6Zad) -+
u,
561
Lepton
M
+ e + D ~ =));(
1, one can write for the branching
):(
5
BR(p--$u,,+e+v,).
(16.7a) Using the experiment,al values, we get,
B R (T
-+
uT
+ e + De) = (17.82 f0.09)%
(16.7b)
to be compared with the experimental average (17.83 f 0.08)%. If we neglect mJm, , we get for the decay T -+ u, e v,,, the same expressions as in Eqs. (2), (3)) (4) and (6). However, taking into account the finite value of mJm, we get,
+ +
where
K(y) = 1 - 8y
+ 8y3 - y4 - 12g2In y.
Using the experimental values of m,, and m, (8) and (4)
B R (T
-+
V,
+ p + 0,)
(16.9)
, we get, from Eqs.
(0.9726 f 0.0001)BR(~ uT = (17.33 f 0.09)% -+
+ e + De) (16.10)
to be compared with the experimental average (17.35310.1)%. Thus we see that, e - p - T universality is satisfied to an excellent, degree of accuracy.
Weak Decays of Heavy Flavors
562
16.2 Semi-Hadronic Decays of 7 Lepton We consider a general decay
where X is any number of hadrons allowed by energy conservation. The T-matrix is given by
T = --G’ (0
a
IJ,”IX) a(k’)y”(l-
75)
LdZ.
u(k)(243
(16.11)
The decay rate is given by
where
Note that G’ is the effective decay constant, LPAis the leptonic part given by
P ’ Since the interference term The weak current J,” = VCLw - AW VPA, does not contribute, we can separately consider the vector and axial vector parts. Using the Lorentz invariance and CVC (the spin over final hadrons is summed), we can write quite generally
0) d3Px6 ( P x - Q)
(16.15)
Semi-Hadronic Decays of
T
563
Lepton
Similarly we can write
(W3/(o = 6 (q0)
x)(x(Ax"+(0) d3PxS (Px - 4 )
IA;1"I
[( - 8 g p A
-k qp4A) PA
(8)-k qp4AaA
(q2)]
(16.16)
In writing Eq. (16), we have not used the conservation of axial vector current. The form factor a A ( q 2 ) arises due to non-conservation of A,. From Eqs. (12), (13), (15) and (16), we get
where s = q2 =
(k + k 1 ) 2 .
(16.19)
Special Cases:
r-
1.
+ IT-
+ u,
Here PA(S) = P V ( S ) = 0,
CA(S)
= f: S(S - m:),
(16.20)
GI2 = G$ cos2 Oc. f T is the pion decay constant and is defined by the matrix element
(16.21)
Hence from Eq. (lg), we get (16.22)
Weak Decays of Heavy Flavors
564
2.
7-
p-
+ u,
--i
Here PV(S)
&f
=
(16.23)
- m;,,
where f, is defined by (16.24) Hence from Eq. (17), we get,
(16.25)
3.
7- -+
a,
+ v,
Here PA(S) =
(16.26)
f:16(s - m i 1 )
Hence from Eq. (IS), we get (16.27) Let, 11s compare these results with their experimental values. From Eqs. (22), (25) and (27), we have _ r7r -- (127r2)
re
5
=
(5)
cos2 8, (1
(12,,,(-5),,,,(,-3
$)
2
-
2
2
(16.28) (1+2$).
re
(16.29)
re
=
(122) ( ~ ) c o s ~ 8 , ( l - ~ ) 2 ( l + 2 ~ ) (16.30)
Semi-Hadronic Decays of
T
565
Lepton
Using f n = 132 MeV, cosB, = 0.97 and the experimental values for masses, we get
BR(T- t v7r-) M (0.605)
BR(Y
+ v,p-
)
=
(
-
i2)'
Be
(16.31a) (24.94 & 0.16)%
(0.557) (&)'Be \
(11.31 f 0.15)%
,
(16.31b)
BR(T- -+ v ~ + u = ~ )(0.329)
Be
(17.65&0.32)%.
(16.31~) Using B e = (17.83f0.08)%, we see that to get the experimental values given in the parentheses in Eqs. (31), we should use
ful
MeV. (16.32) We now show that the above values of the decay constants firand f p as extracted from -r decay are consistent with those determined from light flavors physics showing the inner consistency of the standard model. To see this we first note that the K S R F relation and the Weinberg first sum rule give respectively
fir= 134 MeV,
fp
= 208 MeV,
On the other hand the decay width for po bY
r (p
t
e+e-) =
--t
= 229
ese-
is given
(16.34)
The comparison with its experimental value 6.77 keV gives f p M 216 MeV in agreement with that in Eq. (32) and the latter value is also consistent with the K S R F value f,, M 190 MeV.
Weak Decays of Heavy Flavors
566
On the other hand the Weinburg sum rule gives MeV, incompatible with that in Eq. (32).
fol M
130
One can improve the theoretical predictions of r(T- -+ p-v, -+ lr-lrov,) and r(7- -+ a;v, -+ lr-pov,) by taking into account, finite decay widths of p and al mesons (see problems 1 and 2). However, for hadronic decays ( T - -+ f- v,) which proceed through the vector current only, one can use CVC to relate r(7- + f- v T )to the scattering cross section for the process:
+
+
-+ y
e-e+
-+
fo.
The cross section of this process is given by Eq. (A.79) 167r3a2 crf (s) = ___ Pr S
(4
(16.35)
Using CVC, we get, (16.36) Hence we have
x
/" (1 4) (1 + s)
s af(s)ds. (16.37)
-
mT
mT
Let us apply this to the decay 7
-
0 0 0
7r lr
-4
lr-lr
--t
7r-7r-7r+7ro
Then we get, from Eq. (37)
+v, + v,.
(16.38)
567
Weak Decays of Heavy Flavors
Since the decay proceeds via I = 1 weak currents, one also obtains an additional relation
r (.-
-+x - 3 x 0 4
=
(16.40) We have discussed above the dominant decay modes of T viz. T - ---t v, e-v,, 7- + vTpL-vP,T - + v , ( ~ T ) - , T - + v7(3x)and T - t v7(4n)-. The other small decay modes can also be estimated. All these decay rates occur at the expected rates. The agreement between the theoretical and experimental values is good for each exclusive decay mode. Finally if we add the decay rates for all exclusive channels for 1 prong events [T- + V, (particle)- neutrals (2 O)], we get the value (84.96 f0.14)% to be compared with direct inclusive one prong branching ratio B1 = (85.53 f 0.14)%. Thus there is no discrepancy between the two branching ratios. 7-decays are well understood in the standard model.
16.3 Weak Decays of Heavy Flavors In the standard model, the hadronic charged weak current, can be written as (see Chap. 13)
JW P
=
(nz Z)yp (1 - 75) v
( ;) ,
(16.41)
where (16.42) h d
&s
&b
is the Cabibbc-Kobayashi-Maskawa (CKM) matrix. As we have discussed in Chap.13, the matrix V has only four real parameters [see Eq. (13.156)].
568
Weak Decays of Heavy Flavors
Since ll&l << [I&[, for the Cabbibo favored decays of the charmed mesons (c -+ s + W') , we have the selection rule:
Ad) = A C = A S
(16.43a)
whereas the decays with
AQ=AC,
AS=O
(16.43b)
are suppressed. The decays for which
Ad) = A C = -AS
(16.434
are strictly forbidden in the lowest, order. Thus we expect, D mesons (ca, cd) to decay predominantly into states wit,h stxangeness S = -1 ( K - + anything) and Ds xncsom (cS) to decay predominantly into states with strangeness S = 0 D, + 4 (7r's)' , K*+E(,
[
K*OK+, (.,S)+].
Since IV,bl << IL'&l, for the Cabibbo favored decay of B mesons (6 C W + ) ,we have the selection rule --f
+
AQ = A B
= AC
(16.444
whereas the decays with
AQ=AB,
nc=o
(16.44b)
are suppressed. The decays for which
AQ
= AB =
-OC
( 16 . 4 4 ~ )
are strictly forbidden in the lowest, order. Thus we expect, B (ub, db) meson to decay predominantly into states with C = -1 and B, (sb) to decay predominantly into states with C = -1, = -1.
s
Weak Decays of Heavy Flavors
569
16.3.1 Leptonic decays of D and B mesons The decay constants fD, fD,, fs, fs, can in principle be determined from the leptonic decays of D and B mesons. Thus for example, using Eq. (41), we can write
(16.45a) where
In order to determine fo,we need IV,dJ2 and llLd12 . Now IV,dI2 can be determined with a great, degree of accuracy from niiclear P-decay. Its value is given by [ see Chap. 111. IV,dl
= 0.9750 f 0.0007.
l K d l has been determined from
(16.46a)
u and 0 production of charm in
deep inelastic scattering. Its value is
J&J = 0.221 f 0.003. Using IVudl M 0.97, lVcdl = 0.22, and imental values
+ up) -+p+ + up)
(r+ -+p+
F (D'
fT
(16.4613)
132 MeV and the exper-
=
2.53 x
MeV,
< 4.48 x
MeV,
=
we get,
fo < 288 MeV.
(16.47)
Needless to say we can write similar expressions for the leptonic decays of D$, B* and B$. For 0," decay,
llLs1= 0.9743 f 0.0007
(16.48a)
570
Weak Decays of Heavy Flavors
and
( D t -+ p+
+ u p ) = (13 f 6) x lo-’’
MeV,
(16.4810)
so that, from Eq. (45), we get,
247 MeV < f ~ <, 406 MeV.
(16.48~)
For B+ 4 p L + v pwe , can express the branching ratio:
B R ( B + + p+vp)
(b) (&/’. fn 0.003
= [1.27
(16.49)
Thus one can directly determine fB IVubl from this branching ratio whenever the experimental data are available. 16.3.2 Semi-leptonic decays of D and B mesons The prototype of these decays is Ke3 decay ( K - -+ no eUe). In fact from this decay and hyperon decays, lVu81has been determined: lVusl = 0.2205 & 0.0018. (16.50)
+ +
It is theoretically simplest, to begin with semileptonic decays of heavy flavors. We start with the decay D- + X o + e-
+ ge :
p =px
+ kl + k2
where X o is any number of hadrons consistent with energy conservation and allowed by the selection rules. The T-matrix for this decay is given by
T
= -~ GF Ks
Jz
x
(x~ j D> r /L / ! G G
[W)YP
(2 .)3
h o
(1 - Y5) 21 (k2)l.
k20
(16.51)
Since experimentally, we observe only charged leptons, we sum over hadrons. Thus for the inclusive semileptonic decays, we get for the decay rat,e:
r=
(mem,) Lpx A,x
(16.52a)
Weak Decays of Heavy Flavors
571
where
Here Q = ( ki
+ k2) ,
q2 = 2EeE, ( 1 - cos 0)
(16.53) Note in writing Eq. ( 5 2 ) , we have neglected those form factors which give contribution proportional to lepton mass (me).In Eq. ( 5 3 ) , we have also put rn, = rn, = 0, and kl0 = Ee and kzO = E,. Hence we get from Eqs. ( 5 2 ) and ( 5 3 )
dr d E e d u dQ
+-4 1
m D
(Ee - E,)
1
5 ( 1 - cos8)
f3
(u,q2)] .
(16.54)
This is a general expression for the semileptonic decay rate of D meson. But this expression is not useful since we do not know
572
Weak Decays of Heavy Flavors
C
S
P
P'
Figure 1 Dominant (spectator) diagram for Cabibbo favored semilep.tonic decays of D mesons.
the form factors f l , f 2 and f 3 . In order to determine these form factors one has to use some models. The simplest, model is the spectator quark model. According to trhismodel the decay proceeds as shown in Fig. 1. It is assumed in t,his model that, the quarks in the final states fragment into hadrons with iiriit, probability. In this model, one can easily calculate t,he tensor AILxusing the iisiial trace techniques. Noting that, P = x p , P' = P - y, we get,
ApA
= 27-S (2x)4
(274
1
( 2 % ~y. - rn,2
XPO
x [2x2 p, PA -
+ rn,2 - y2)
+ 9xp (-4 + X P
'
Y)
4.
x i&,xaa P"
(16.55)
Thus comparing it, with Eq. (52), we obtain f2
(v,y2)
=
s 2xp. y - m, + m, "
4
fl (w2)= -; (-4 + xp f3
(v,y2)
=
2
82 rn,
*
4)
- Y2)
s ( 2 x p . q - rn,2 + rn? - qz)
-8 rn; 6 (22p - q - m,
2
+ rnp - q')
.
(16.56)
573
Weak Decays of Heavy Flavors
Hence from Eq. (54) and noting here that
IC
= mo/mcM 1, we get
(16.57) where
(16.58) Equations (57) are exactly the same as one gets (see problem 11.1 with G$ replaced by Gg IvC3I2 ) for the decay
c
-+
s + e-
+ ve (c -+ s + e+ + ve) .
Similarly for the (inclusive) semileptonic decay of B mesons viz.
B+ -+ XO
+ e+ + ve ( B -
--f
XO
+ e- + D ~ ) ,
the basic process is
6 c se+ + v e 4
( b + c + e-
+ ue)
and we get (see problem 11.1with G$ replaced by G$
IvhCl2).
(16.59)
Weak Decays of Heavy Flavors
574
where (16.60) Note that the difference between Eqs. (57) and (59) is due to the fact, that V - A interference term [the third term in' Eq. (54)) has opposite sign in the two cases. We conclude this section with the remarks that according to this picture, we get
r (D+ + XOe+v, ) r (B' + Xoe+ve ) For the decay
D:
--+
=
=
r (DO r (Bo
-+
X-e+v,
--+
X-e'v,
) ).
(16.61)
x0+ e+ + v,
we get for $ and F exactly the same expressions as those given in Eq. (57). Similar remarks are applicable to the decay Bt -+ X- e+ v,. Hence the quark model predicts
+ +
r (D+
Xoe+ve)
r (DO
) = r (D:
Xoe+ve), (16.62) r ( B + --+ Xoe+v, ) = r ( B O --+ X-e+ve ) = r (BY --+ X - e + v e ) . (16.63) The experimental branching ratios for these decays are: --+
=
--+
(D+)sL
=
B R (D'
-+
DO)^,,
E
BR (DO
-+
( D " + ~f ~ B R (D,+--+
X-e+v,
-+
) = (17.2 f 1.9) % X-e+ve ) = (7.7 f 1.2) % (16.64) Xoe+v, ) < 20 %. Xoe+ve
For B mesons t,he experimental values are
(B+)sL
= =
B R (B'
--+
X o e f v , ) = (10.1 f 2.3) %
B R ( B o--+ X-e'v,
Now the experimental values for respectively (1.057 f 0.015) x
) = (10.3 f 1) %. (16.65)
T D + , T D O , T ~ , + TB+ ,
(0.415 k 0.004) x
and
TED are
(0.467
575
Weak Decays of Heavy Flavors
f0.017) x 10-l2, (1.62 f 0.06) x lo-", (1.56 f 0.06) x Thus we see that T ~ Ox # TD+, TD+ M 2.5 TOO i.e. isospin is badly broken for D+ and Do. Also we note that TB+ M TBO. Thus one would expect using Eq. (62)
This is consistent with the experimental value given in Eq. (64). We can conclude that Eqs. (62) and (63) are well verified experiment ally. In the end, we note from Eq. (57) that we can get, an estimate of [K31,using the B R ( D Xev,) from the experiment!if we know the quark masses m, and m,. 16.3.3 (Exclusive) semileptonic decays of D and B mesons Our general formulation can be used to calculate the decay rate for semileptonic decays of the type ---f
where P = D or B. M is a pseudoscalar meson P' or a vector meson V . In order to discuss these decays, we first define the form factors
(16.67a) where
t
= q2 = ( p - p / ) 2 ,
Weak Decays of Heavy Flavors
576
Here mp is the mass of P and mpr is the mass of PI. For the vector meson V, the form factors are defined:
(16.68a)
(16.6813)
”
(16.69) However for the transition B -+ D or B -+ D*, the heavy quark spin symmetry gives the following relations among the form factors [cf. Eqs. (9.42), (9.58) and (9.59)]
Weak Decays of Heavy Flavors
= A2
-
577
( t )= A+ ( t )= -A- ( t ) = -A0 ( t ) (16.71a)
A1 ( t ) = A ( t ) =
mi3
+
[l
mD*
+ 21
*
21'1
(u *
21')
(16.71b)
Note that
t
=
2 mB
2 + rnD* - 2mB rnD* u - u'
2 = mB t m,.2
-2
m mD. ~ w
(16.72)
where w=u*u'
At w = 1, t = normalized as
(mB - r n p )2 --
( (1) = 1,
t,,
(16.73) the form factor ((w)is
t ( ~ m a x )= 1.
(16.74)
Having defined the matrix elements arid form factors, we give in the following table the exclusive semileptonic decays, which we are considering:
Weak Decays of Heavy Flavors
578
Now for the semileptonic decay P -i A4 Zv, since the meson M is on the mass shell, we can integrate our master equation (54) over Y using the delta function 6 (2m,v - m; - t mb).We obtain
+
G2 1 [4- Ee - t ] f2 ( t ) 2t fl (t) dt dEe ( 2 4 3 SmZ, t (16.75) [2Ee f 3 (t)} , mP where K is the momentum of meson A4 in the rest frame of P , t = q2 and G = G F I V q ~ IIntegrating . Eq. (75) over the lepton energy Ee we get, d r
- -
+
{
-
mT]
+-
Note that in Eqs. (75) and (76), we have neglected those form factors which give contribution proportional to lepton mass rnl. First we consider the case when A4 is a pseudoscalar meson i.e. M = I".In this case we get, from Eqs. (67) and (52c) f2
(t)= 4
4 IF+@)I2
,
fl ( t )= 0.
(16.77)
Hence (16.78) For the vector case i.e. A4 = V, we get from Eqs. (68), (69) and (52c) after straightforward but some what, lengthy calculation
Weak Decays of Heavy Flavors
579
where we have used
KV = mv d m , Ev = mv w 2 t = m p mv 2 - 2mpmv w
+
(16.80) (16.81)
For the transition B
--t
D*,it is convenient to define
R l ( t ) = [1 Rz(t) = [l -
(16.82)
Note that in the heavy quark symmetry limit R1 ( t ) = R2 ( t )= 1. Using Eqs. (81) and (82), we get from Eq. (79) with mp mB, mV ---t mD*, --f
+ [(m; + m;.
- 2mB m,.)
4w (1 + w + l
-)]
x R: ( t ) }F2(w).
(16.83)
In the symmetry limit, R1 ( t )= R2 ( t )= 1, and we obtain
(16.84)
Weak Decays of Heavy Flavors
580
For B
dw
-+
-
D transition, we obtain from Eqs. (78) and (70) G; lKb12 ( r n B + r nD)2 rn; (w2 - 1)3'2 G2 (w) . (16.85) 48r3
In the heavy quark symmetxy limit,
s (w)= .F (w)= c
('w)
.
(16.86)
Let, 11s first, consider the semileptonic decays of D-mesons. The experimental results on the form fact,ors are as follows:
D+ -+ K- 0 !t + ve D+
'
-+
:
F+ (0) = 0.74 f 0.03
K*ol+ve:
(16.87)
V (0) = 1 . 0 f 0 . 3
A1 (0) = 0.55 f 0 . 0 3 A2 (0) = 0.40 f 0.08 ~2 (0) = 0.73 f 0.15 (16.88)
rv (0) = 1 . 9 0 6 0.25
v
Ds+ -+ @+V, : (0) = 0.9 h 0 . 3 A1 (0) = 0.62 f 0 . 0 6 A2 (0) = 1.0 f 0.3 ~2 (0) = 1.6 f 0 . 4 TV (0) = 1.5 f 0.5
(16.89)
These form factors are of import,ance for t>wo-bodynonlept,onic decays of D-mesons in the factorization ansatz (see the next, section). There is no model independent way to determine these form factors. First we note that the form factors for various decays are related by SU(3) as follows
-&F+
(D+--+r0)
=
F+ ( D 0 - + C )
Weak Decays of Heavy Flavors
For D
581 =
F+ ( D + + K O )
=
F+ (DO --+ K - )
=
-mF+ (Df
=
F+ (DS++KO).
+%3)
V transitions, we have similar relations with
(16.90)
and K 4 K* and since the nonet, symmetxy holds for vector mesons, we have ---t
v
( D L p - )
= = =
=
and
v
( D t +J)
7r + p
v (D+4*o) v (DO+K*-) v (D;+K*O) -v (Dg*-+$) = 0.
(16.91)
Needless to say that similar relations hold for axial vector form factors A1 and AZ. Most of the models agree that the form factors F+ and V are dominated by vector bosons D* and 0:. Hence for the Cabbibo favored decays Do + K-!+v, K * T v and 0,' --+ $ P v , the vector mesons dominance shown in Fig. 2 gives
(16.92)
Weak Decays of Heavy Flavors
582
Figure 2
Vector meson dominance for two-body decays of
where we have parametrized gD:oKand
gD:DK'
Do
as follows (16.94) (16.95)
+
For AD = 1, Eq. (94) follows from SU(4) (flavor spin) current, algebra and Eq. (95) with A D = 1 is also a consequence of SU(4) algebra, vector meson dominance and the K S R F relation. Using SU(3), we can write
Note that all the form factors for Di --+ q5 have negative sign relative to Do -, K - , so that an overall minus sign in front of Eq. (96) can be ignored. Comparing Eqs. (92), (93) and (96) with Eqs. (87), (88) and (89), we have
X D f-DL* fK
=
0.74 f 0.03
(16.97)
Weak Decays of Heavy Flavors m D AD
583
+ mK' s fD* 1.0 f 0 . 3
(16.98)
fK
mD;
(16.99) mDb+mg = 1.41, f~ = 1.25 fir, fs = Now using m DmD: + m K * = 1.30, mD: 1.2 f n , we get from Eqs. (98) and (99) respectively
AD-
f 0:
- 0.8 f 0.2
(16.100)
fK fD: ADfK
fs
= - (0.6 f 2)
z (0.6 f 2)
I
(16.101)
fK
Thus within the experimental errors, Eqs. (97)-(99) are consistent with each other. On the other hand, if we use f D : = 275 MeV, fn = 132 MeV, f K /fir = 1.25, we get from Eq. (97)
However, since fD: is also not, well determined, it, is safe to say that AD is of order 1/2. The value of AD can be determined from the experimental value of the decay width for the decay D*+ -+ DOT+. The decay width is given by
r (D*+ D%+)
=
gD'D?r P3 67r m&,
A; (181) keV = 45 keV, =
(16.102)
for AD = 1/2. Experimental upper limit on r is I'(D*' + DOT+) < 89 MeV. With improved experimental numbers in Eqs. (87)-(89) and for I?, better information on these form factors can be obtained. For B --t D , D* transitions, the heavy quark spin symmetry gives R1 ( t ) = R2 ( t ) = 1; but it, does not give the form factors
Weak Decays of Heavy Flavors
584
F ( w ) and
(w) at any w except, at, w = 1. Taking into account, symmetry breaking corrections to the heavy quark limit,, it is found
F(1) = 0 . 9 2 4 f 0 . 2 7 S(1) = 1 . 0 0 f 0 . 0 7 .
(16.103)
A more refined analysis of symmetry breaking gives
Rz
M
I--.
A
(16.104)
2%
If we use, a3(m,) M 0.34, A M 0.41 GcV, [cf. Eq. (9.91)] m, M 1.5 GeV we obtain R1 M 1.3, Rz M 0.9. The data can be fitted by assuming R1 and Rz as constants and by writing
3(w)= F ( 1 ) [ l - p i 1 (w - l)]
1
(16.105)
The fit, t,o the data gives
R1
1.18f0.30f0.12 Rz = 0.71 f 0.22 f 0.07 PA1 = 0.91 f 0 . 1 5 f0.06. =
(16.106)
Thus we see that the values of R1 and Rz are in good agreement, with the predictions of HQET given in Eq. (104). I21 and Rz arc rather insensitive to t,he form assiimetl for 3 (w).However, thr: value of p i , is sensitive to the form of 3 ( w ) . 16.3.4 Non-leptonic deca,ys of B and D mesons At tree level, the AB = 1 non-leptonic weak decays are described by a single W-exchange as shown in Fig. 3, which represents the decay b --t u + q’ + tj ( q = u or c; q’ = d or s) . Here a and ,8 arc color indices. The effective Lagrangian for this decay is given by
Weak Decays of Heavy Flavors
585
y*. u
A Figure
q2
Decay o
4
C(U)
+ q' +
through W-exc,range
Since quarks carry color, the QCD corrections must, be taken into account. Under QCD renormalization, the Lagrangian (107) becomes [see Fig. 41:
Leff =
% [&
&b
+
vub
v ; q l
(clo?+ c20;)
VGl (cloy+ CZO;)
q=u,c
]
(16.108)
where Ca are Wilson coefficients evaluated at the renormalization scale p; the current-current operators 01,~ are
and 0: are obtained through replacing c by u. Here
(c" b p ) V - A
= c" yp (1 - 75) bp etc.
For c -+ s + u + 4 ( q = d or s) , replace b by c, c by s and q' by u. Note that strong interaction due to hard gliion corrections has been
586
Weak Decays of Heavy Flavors
Figure 4 QCD corrections to the Fig. 3, 0 ’ s denote standard operator insertions
taken into account in the Wilson coefficients C1 (p) and Cz ( p ); without these corrections C1 = 1 and CZ= 0. The long range QCD effects are taken into account by the matrix elements of operators 01,2 between hadronic states. They manifest themselves in the form factors. To the leading logarithmic approximation (LLA),the Wilson coefficients Ck = C1 f C, [note that QCD corrections mix the operators O1 and 0 2 , requiring diagonalization which result in O* = 01:02, C* = C1 f Cz] are given by [see Appendix B] (16.110a) where C* (mw) = 1 (in the approximation we are using) and y* = yi/2,&, with y i = 33so that, for N , = 3 [PO= 4 ~ b ]
and n f is the number of active flavors (in the region between p and m w ) . At p = m b = 4.9 GeV we get from Eqs. (110), taking a, ( m w ) M 0.12, n,f = 5 and a, ( m b ) = 0.22, as (m,) = 0.34,
c]( m b )
1.11,
c 2 (mb) !Z
-0.26.
(16.111)
Weak Decays of Heavy Flavors
Figure 5
587
QCD penguin diagram
They are not very much different, from the Wilson coefficients at p = 2.5 GeV in next-to-leading logarithmic ( N L L ) precision:
C l = 1.117,
Cz = -0.257.
(16.112)
At p = m,, we get from Eqs. (110a)
Ci (m,) 1.24,
Cz (m,) G -0.48.
(16.113)
In addition to the current-current operators Oi(i = 1,2) in Eq. (108), originating in the usual W-exchange and subsequent QCD corrections, there are QCD penguin operators O,(i= 3 - 6) originating in the QCD penguin diagrams shown in Fig. 5 . These operators add to the effective Lagrangian (108) the following term 6
-&b
&*, xci i=3
where
oi
(16.114)
W:ak Decays of Heavy Flavors
588
However, the Wilson coefficients here are miich smaller than C1 and C2 :
C,= 0.017,
Cs = 0.011,
C4 = -0.044,
C,= -0.056.
(16.116) Thus generally we shall not consider the penguin operators. Now for the Lagrangian (108), or the corresponding one for c -+ s + u + Q, it, is simple to write the decay widths for D , B -+ hadrons in the spectator quark model for t,he Cabbibo favored decays:
r [ D (c 4) =
---f
q
((J
s
+ d + 41
r(c-ts+d+u)
(16.117)
=
r(
b-+c+d+ii -+c+s+c
-
+0.12
r
+c
+d +u]
(16.118)
where we have iised (16.119) We also take
B ( b ---t x 7 B (b 3 X c
-
u ~ M) 0.24. ve)
(16.120)
Weak Decays of Heavy Flavors
589
Hence from Eqs. (62), (117) and (63), (118), (119) and (120) we get
M
16 %
(16.121a) (16.121b)
and
(16.122a)
rgo -
- rB-
1
(16.122b)
where in Eqs. (121) and (122), we have used the values of C l and Cz given in Eqs. (113) and (112) respectively. Let 11s first discuss ( D ) s L, while Eq. (121) is consistent, with the experimental value for but, it, is about, a factor of two greater than (DO),, . Also we note that the experimental values, namely (16.123) FLY PD+
-
-- - 2.55 f0.04 rD+ TDO
(16.124)
are in disagreement with the predictions of spectator quark model given in Eqs. (121). A possible explanation is as follows: There are two spectator diagrams shown in Fig. 6, similar to Fig. 3 for B decays. Figures 6a and 6b correspond to the charged and neutral current operators in the Lagrangian (108) and are multiplied by the coefficients C1 and Cz respectively. For D+, ij = d , and in the two diagrams we have the same final states (for example K- 0 7r + ). The two diagrams destructively interfere, so that r (D+ ---t hadron) o(
590
Weak Decays of Heavy Flavors
Figure 6 Spectator diagrams corresponding t o the effective (a) charged (b) neutral current operators.
+
3 (C, CZ)’ . On the other hand for D o , 4 = ii and we have different final states for Figs. 6a and 6b (for example K-n+ for Fig. 6a and Koro for Fig. 6b). Thus in this case r ( D o --+ hadrons) cx 3 (C,” C,”). Hence we have
+
(16.125b) Thus whereas Eq. (125b) is in agreement. with Eq. (123), Eq. (125a) has still some discrepancy. Numerically
(D+)sL M 27 %,
(Do)sL
14 %.
(16.126)
Both these values are not, in agreement, wit,h their experimental values given in Eq. (64). The spectator approadh is expected to be a better approximation for B decay as b quark is heavy. We now discuss the comparison of the prediction of spectator model for the semi-leptonic branching ratio BsL with its experimental value. First we note that the naive quark model gives BsL M 15 % [cf Eq. 122al. Charm
Weak Decays of Heavy Flavors
rP
59 1
< 7 c j' S
P;
Figure 7 W-exchange quark level diagram for hadronic Do decay. mass corrections to r ( b -+ ccs) have been found to be large and can reduce the theoretical prediction for Bsr, :
Bsr, = (11.7 f 1.4 f 1.0) %
(16.127)
The CLEO and Argus collaboration have measured the branching ratio BsL : Bsr, = (10.3 f 0.39) %. (16.128) The corresponding LEP number measured from the Zo + b6 decay rate are consistent with (128). There does not appear to be any significant discrepancy between Eqs. (127) and (128). 16.3.5 Scattering and annihilation diagrams There are two kinds of mechanism for the hadronic decays of heavy mesons which we have not, considered. They are depicted for Cabibbo favored decays of Do and D, mesons in Figs. 7 and 8 respectively. The basic processes depicted in Figs. 7 and 8 are respectively c + fi + s + d and c + 3 -+ u + d where for the second process, the analogue of Eq. (108) gives the effective Lagrangian
Weak Decays of Heavy Flavors
592
Figure 8 W-annihilation quark level diagram for hadronic Df decay. where (‘ = lKs1 IT/;ldI while for the first, process the effective Larigrangiari is obtained by Fierz rearrangement. Thus the T-matrix for the process c u + s d is given by
+
+
where we expressed t,he wspinor in terms of u-spinor by the relation 7) = Ctu”. Sirriilarly for t,he aririihilat,iori diagram (Fig. 8), Eq. (129) gives thc T-mat,rix:
where we have used the Fierz rearrarigcrrierit, in going from the first, line to the second line. Thus we co~icliidethat, bot,li diagrams
Weak Decays of Heavy Flavors
593
give the same results apart from the color factors. In taking the nonrelativistic limit it is convenient to express v-spinor in terms of u-spinor and use the relation 17 Pi v = ci u ri 21, with E~ = f l , ri= ?A, 7 ~ 7We ~ .note that mi, mg = m (the mass of u and d quark), mi = m, >> m, mj = m, or md; E,! = Ei = E , m, Ej = 2E, where in the nonrelativistic limit, we have piit, Ei = m, and we also put, ma= md. Using Paiili representation of Dirac matrices, it, is a straightforward but long calculation to obtain the cross section CT for the scattering or annihilation processes shown in Figs. 7 and 8. Suppressing the color factor, we get, for the singlet, and triplet, scattering cross sections respectively
+
IEI2
as = -G; 81r gT
=
1
(8m2)-,
1tI2
-G; 8n
(16.131a)
91
1
(:m:)- PI
( 16.131b)
where is the incoming velocity in the initial state. Now defining the decay width as r = p3(o)12 u, (16.132) we get for the triplet state 1 8 r ("SI -+ u d-) = G i E2 - m& 8n 3
19,(0)12,
(16.133)
+
where we have put m;, = (m, ma)' M m;. For the singlet state, we get
('So
+u
1
2) = Gg t 28 m2 8n
(0)l"
(16.134)
Note the important fact that the decay width for the singlet state (D)is proportional to the square of the light, quark masses; in the spectator quark model it is proportional to m:. This is called helicity suppression.
594
Weak Decays of Heavy Flavors
Inserting back the color factors we have finally r,&h =
c% 87r
l%slZ
I V U(8~ mz) ~ ~ I@s ( ) I 2
(CI 4- 3c2)2, (16.135)
r:;n
=
87r
IV..I2 lV,dI2
(8 m 2 )[as (0)12(3C1
+ C2)’. (16.136)
It is clear from Eqs. (135) and (136), that both the exchange and annihilation diagrams are helicity suppressed, but rezch is color suppressed and is color enhanced. It is intresting to see that, for the annihilation diagram, one can get, the same result, just, by writing the T-matrix for the D, ---t hadrons in the form
rann
where J,”t and J W , are color singlet, currents with appropriate quantum numbers. Then
I(xIJW’I
x I(0 I-r,”tI a)I2 Now from Lorentz invariance
O)l2
*
(16.138)
(16.139) while
-
1 (2,)38 (Po) [ (-P2 +P, PA
(7
(P2)] ‘
SPA + P, PA) P (P2)
(16.140)
Weak Decays of Heavy Flavors
595
Hence we get
(16.141) Now from dimensional consideration
(16.142) Thus we obtain
One gets eactly the same results if in Eq. (137) one replaces (X) by Id)(see problem 3). Comparing Eq. (143) with Eq. (136) we get
(16.144) It is interesting to note that the vacuum saturation of the T-matrix for D, -+ hadrons viz. (X !Jw’”(0) (0 IJ,”t( D,) gives the same results as the annihilation diagram. From Eqs. (117), (135) and (136) and (143)’ we get
(16.145)
+
where in Eq. (146) the factor 3 (Ct Ci) appears for the reason discussed earlier in connection with Eq. (125). Using C1 =
596
Weak Decays of Hcavy Flavors
1.24, Cz = -0.48, m, M 1.50 GeV, F(m,/m,) M 0.47, fD = 200 MeV and f ~ =, 240 MeV, we get from Eq. (144)
( 16.147) ~~ Thc annihilation diagram gives negwhile I ? ~ ~ , J Iis' negligible. ligible contxibution to D s decays. Taking into account, Eq. (147)
r ( D s + hadrons) = (1.6) rSP where
rspis giver1 in Eq.
(16.148)
(117) and [cf. Eq. (121)]
+ c;)
3 (C? M 0.82. (16.150) 1.6 [3C? 2C1 Cz 3Cg] 7110 While the prediction (149) is consistent, with the experimental limit (D:),, < 20 %, the prediction (150) is not consistent, with its experimental value M 1.12. We conclude that, the contribution of the annihilation diagram is helicity suppressed, but enhancement by a factor of 1 9 2 ~ ' due to phase space and that due to color factor more than compensate the helicity slippression. However, there are still problems to explain the ratios rDt/rDoM 2.5 and I - ~ ~ , + / T ~ O M 1.12 as was discussed in Sec. 3.4. Finally the aririihilation diagram for B decays are Cabibbo suppressed and they may be neglected. 2 To i~ N
+
+
Problems
597
16.4 Problems 1. Taking into account finite width for p meson and using Eq. (17), show that
where
Hint: 2nS (s - m;)
-+
(s - ma)
+ m: r2
Considering the process e-e+
--+
y
--f
n+n-,
show that the cross section is given by (s >> 4m:) or+*- ( s ) = n a2
3s
I F,
( .)I2 (1 -
!33'2
where F, ( s ) is the electromagnetic form factor of pion:
Using Eq. (37), show that we get, back Eq. (A). From Eq. (A), find the decay rate for T- -+ n-7r0 vr through p-resonance.
Weak Decays of Heavy Flavors
598
2. Taking into account, finite width of al meson, and Eq. (18), shdw that, (taking m, = 0)
where F p x (s) =
fa, Faipn
[(s - m&) + a ma,
r].
The a l f n couplings are defined by the decay amplitude T:
where qp, E~ are polarization vectors of a1 and p, q and k are their four momenta. In order to derive (B), first show that
599
Problems
Using the experimental numbers for I‘( T - + T - po v7) and r ( a l Falp,and using = f,”= 2F; m;. 3. Using Eqs. (137) and (138) and writing + p T ) , determine
/ d3 Px 8 (P
T-)
- Px> -+
/
d3
Pl d3 P2 6 ( p - Pl - p 2 ) ,
show that
4. Writing ( O I J pwt
I D1)
f D : &p
where cP is the polarization of D;, show tjhat,,
Comparing it with Eq. (133) when multiplied by the color factor (3Cl C2)2,show that,
+
j& Hence show that
=
12 19, (o)12mp.
Weak Decays of Heavy Flavors
600
16.5 Bibliography 1. M. L. Perl, Rep. Prog. Phys. 55,653 (1992). 2. A. S. Schwarz, T physics in “Lepton and Photon Interaction” XVI Int. symposium, Ithaca NY 1993 (eds P. Drell and D. Rubin) AIP, p. 671; M. S. Witherell, Charm decay physics, ibid p. 198; M. B. Wise, Heavy flavor theory; ibid p. 253 3. Particle Data Group, Eur. J. Phys. C, 3 (1998). 4. J . L. Rosner “B Physics - A Theoretical Overview’’ Nuclear Instruments and Methods in Physics Research A 408, 308 (1998). 5. M. Neubrat, “B Decays and the Heavy-Qiiark Expansion” CERN-TH/97-24 hep-ph/ 9702375 [ To appear in the second edition of “Heavy Flavors” edited by A. ,J. Buras and M.Lindner; World Scientific] “Heavy-Quark Effective Theory and Weak Matrix Elements” CERN-TH/98-2 hep-ph/ 980 1269 [Invited talk presented at Int. Europhysics Conf. on High Energy Physics Jerusalem], Israel 19-26 Aug. 1997. 6. M. Neubrat, and B. Stech, “Non-Leptonic Weak Decays of 13 Mesons” CERN-TH/97-99 hep-ph/ 9705292 [ To appear in the second edition of “Heavy Flavors” edited by A. J. Buras and M.Lindner; World Scientific] 7. G. Buchalla and A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996).
Chapter 17 GRAND UNIFICATION, SUPERSYMMETRY AND STRINGS 17.1
Grand Unification
As we have seen all fundamental forces are of gauge nature. Thus they may be deduced from some generalized gauge principle. Ingradients of gauge models are (i) Choice of gauge group (ii) Choice of fundamental representations (iii) If gauge symmetry is spontaneously broken, choice of Higgs sector which generate mass parameters. Gauge principle restricts the form of interaction. Also gauge model may be renormalizable if its fermion content is such that the model is anomaly free. At low energies we have a spontaneously broken SU(2)xU(1) gauge group for electroweak forces and an exact SUc(3) gauge group for the strong quark-gluon forces. Thus the standard model involves
GI
EE
SU(2) x
U(1) x
ScU(3)
Q2
9'
gs
92
The fermion content of GI for the first generation is
601
9'
Grand Unification, Supersymmetry and Strings
602
Thus we have 15 two-component fermion states per generation. The electroweak part of GI is spontaneously broken
GI
S U ( 2 ) L x U(1) x SUc(3) + Gz
Uem(l) x SUc(3).
Also the experimental data show that
which implies that SUL (2) x U( 1) breaking predominantly occurs only through an SUL(2) Higgs doublet, or doublets. Despite the fact that the above picture is capable of providing a current phenomenological description of all the observed “low energy physics” , many questions given below remain: (i) 3 independent coupling constantx (ii) no charge quantization because of U ( 1) factor (iii) no relation between lepton and quark masses (iv) why are 3 generations identical in representation content but, vastly different in mass ? (v) why is the intergeneration mixing small ? (vi) no principle limiting the number of S U ( 2 ) generations - e , p , 7,. * . .
Grand Unification
603
Could the situation be improved ? Grand unification of electroweak and strong quark-gluon forces answer some of these questions but say nothing about, the generation problem. The basic hypothesis is that there exists a simple group G G 3 G1
SU,5(2) x U(1) x SUc(3)
which is characterized by a single coupling constant and that all interactions are generated by G. Quarks and leptons are in general members of the same multiplets of the group G. Then at some energy scale, G suffers a breakdown to G1:
The rank of G 2 4 since the rank of G1 is 4 and some possibilities for G are (i) G = SU(n)
S U ( n - 3) x U(1) x SUc(3)
e.g. S U ( 5 )
(ii) G
5
SO(n) SO(n - 6) x U(1) x SUc(3)
e.g.
SO(10) 3 SO(4) x U(1) x SUJ3) or
suL(2)x suR(2) There may be intermediate steps before reaching the righthand side. Another possibility is SO(10) 2 S U ( 5 ) x U(1). (iii) Exceptional groups E6
3 SU(3) x S U ( 3 ) x SUC(3)
E7 2 S U ( 6 ) x SUc(3)
Grand Unification, Supersymmetry and Strings
604
(iv) Any semi-simplc groiip G=G’xG”x ...
with an additional reflection symmetry will also do.
17.1.1 Q2 evolution of gauge coupling constants a,nd the grand unajication mass scale At presentJy available energies gs, g2 and g’ are very different. How then can we have G wit,h a single coupling constant, ? This is possible since due to qiiaritum radiative correct,ions g’s are Q2dependent,. Thus if we have a grand unification theory (GUT), there must, be a point, Q2 where g s , g2 and g‘ coincide. To see how this comes aboiit, let, 11s consider the Q2 evolution equation for the effective coupling constant in a general gauge theory [see Appendix B for more details]. da,‘ = h+ . . . , (17.1) d 111Q2 for Q2 > masses of fermions and gaiigc bosons biit, Q2 < M$ and (17.2) For SUc(3),C2(G) = 3, TjljAB
=
Tr
(”;’”2”) --
[number of SUc(3) triplets] (17.3)
where n,f is t,he niimbcr of quark flavors, known to be six. For SUr, (2) in the electroweak group,
[t
x - riiimber of left-handed doiiblots] , (17.4a)
605
Grand Unification
where comes from the fact that we have only left-handed couplings. Thus 1 1 TfS - -6 - ( 2 n f ) . (17.4b) Ts - 2 Ts2 Here 2nf = 12 appears since each generation has one lepton doublet and 3 quark doublets (one for each color). For U(1) group of electroweak
C2 = 0,Tf = 2 f
(fy)2
(17.5a)
I
because each fermion has either left-handed or right-handed coupling. Thus
15 - --nf. 23
d In Q2 da;' - d In Q 2 da;' - d In Q 2
(17.5b)
(17.6) -
-
(-
(17.7)
(-pf)
(17.8)
1 22 - 3 2 nf) > 0 b2, b2 = 4n 3 1 2 bl, bl = < 0. 4n
These renormalization group (RG) equations have solution -l ( Q 2 ) Qi
Hence as Q2 increases 1.
Q,
( Q 2 )decreases
= a;'
( m i ) + biln-.Q 2 m2z
(17.9)
Grand Unification, Supersymmetry and Strings
606
2. a2 ( Q 2 )also decreases but, less rapidly than as (Q2)
3. a1 ( Q 2 )increases. Thus, since a1 < a 2 < a, at available energies, at some Q2 = rn$ ,a,, a2 and a1 should coincide
cia, ( M i ) = Cia2 ( M i ) = C:al ( M i ) = aG',
(17.10)
where Cs, C2 and C1 are group theory numbers (so that, the generators of the group are properly normalized) and are of order 1. For example for SU(5), Ci = C: = Cf = 1. Mx is called the grand unification mass scale at, which one has only one free coilpling constant a ~Since . the gauge coupling constants are supposed to merge into one in GUT, the value of sin2O W , which measures the relative strengths of a1 and a2 at Q2= m i , namely
enters into the determination of a~ and Mx.Whether the three coupling constants meet at a single point Q2 = M i depends on the gauge group G. It may be noted that to include the contribution of Higgs doublet one adds - f n H in the expression given in Eqs. (7) and (8), where n H is the number of Higgs doublets. 17.1.2 General consequences of GUTs The general consequences which one would expect from GUTs are 1. G being simple, the charge operator will be a generator of the group and traceless. So if it acts on any representation of G containing quarks and leptons, it would give some relation between quark and lepton charges (sum of charges in each multiplet = 0) i.e. we would have charge quantization.
2. The fact that quarks and leptons share the same representation(s) of G, there would be relationship between quark and lepton masses.
607
Grand Unification
rn
MX
W
Figure 1 Behavior of as( Q 2 ) ,
a2
(Q') and
a1 ( Q 2 )
Q
2
versus Q 2 .
3. Since quarks and leptons share the same representation(s) of G and since gauge theories contain vector bosons linking all particles in a multiplet, there would in general be some interation changing quarks into leptons, thereby violating baryon charge ( B ) and lepton charge ( L ) conservation. At present energy scale E << EGUTx M x , we have effective B and L conservation but this conservation cannot be exact.
In general B violating forces will make proton unstable and so one has to watch that protons do not decay too quickly, the present experimental limit on proton decay is T~
2 1.6 x
years (independent of modes) > 1031 t,o 5 x years (mode dependent,). (17.12)
Using a3 ( m z ) and a ( m ~in) Ms renormalization scheme adopted for the definition of the coupling constants: 03 0-l
(mz) =
0,( m z ) = 0.1214 & 0.0031
( m z ) = 127.88 f 0.09
(17.13)
Grand Unification, Supersymmetry and Strings
608
as inputs, one can predict, bot,h M x and sin2Ow ( m z ) [cf. Eqs.(lO) and (11) and RG equations] in GUT models such as SU(5) with no extra scales bet,ween t,hc electxoweak scale and the GUT scale Mx.Typical predictions are sin28w = 0.215 f 0.003 and Mx M (2
f'
)
x 10''
GeV. This valiie of A4x in tiirn gives ~ ( -+ p
4 x 10"*0.7f1.2 years which contradict,s the experimental limit, ~ ( +p e+n+) 2 5 x years. Likewise the above predicted valiie of sin2 Ow (mz) differ from the present,ly deterrnined value of sin2Ow ( m z ). e+#)
M
sin2Ow
(771%)
= 0.23124 f 0.00017
(17.14)
by six standard derivations. The same mismatch between theory (single - breaking GUT models) and low energy measurements given in Eqs. (13) and (14) is observed if one uses the three effective coupling constants from their measured values to the GUT scale and above. This is shown in Fig. 2, which shows that the three couplings evolved to the GUT scale do not, meet, at, a point,. This observation and t,he others discussed in Sec. 1.2 perhaps point to the presence of new physics between the electroweak scale and the GUT scale. One such candidate is supersymmetry (SUSY), with a SUSY breaking scale somewhere between the electroweak scale and O(1) TeV. One consequence of supersymmetry (see next, section) is that bosonic particles are naturally paired with fermionic ones. Each minimal pairing is called a supermultiplet. For example: a left-handed fermion, its right-handed antiparticle, a complex boson and its conjugate form a chiral supermultiplet. On the other hand a massless vector field and a left-handed fermion form a vector super-multiplet - two transversally polarized vector boson states, plus the left handed fermion and its antiparticle. Thus for n/ = 1 supersymmetry one has the following helicity states chiral: (1/2,0) gauge: (1,1/2) , gravit,on: (2,3/2)
Grand Unification
609
Thus in the minimal siipersymmetxic extension of the st,andard model, we have t,he following particles Particle quark: q lepton: 1 photon: y weak vector boson: W weak vector boson: 2 Higgs: H gluon: G
Spin 1/2 1/2 1 1 1 0 1
Spartner squark: Q slepton: i photino: wino: W zino: higgsino: fi gluino: G
Z
Due to the presence of supersymmetric particles the RG coefficients b’s given in Eqs. (6)-(8)are modified. This modification leads to a solution such that, the couplings do meet, at a point [see Fig. 31. The unification scale in such extensions is higher than the value of M x discussed above in the context, of S U ( 5 ) model. This would imply a longer time for the proton, evading the present experimental bound. Supersymmetry is needed from another point of view, which is discussed in the next section. Before we end this section, we may mention that another popular GUT model, SO(10) [rotation group in ten dimensions in internal space with spinor representations], when broken in a single descent to SUL(2) x U(1) x SUc(3) is also in conflict, with the limit (12). However, in cont,rast, to S U ( 5 ) , SO(10) admits various symmetry breaking patterns, some containing new intermediate mass scales. One such chain of symmetry breaking is
where SUc(4) is the Pati-Salam group. Here it, is possible to avoid the conflict with the limit (12). However, there is no prediction for
610
Grand Unification, Supersymmetry and Strings
Figure 2 Running of the three gauge couplings in minimal SU(5) GUT showing disagreement with a single unification point. [ref. 51
sin2Ow;in fact its value is used to fix the intermediate mass scale mR which is of the order of 1013 GeV and being so large has no observable consequences. To conclude GUTS have several attractive features mentioned above, but their predictive power is limited. However, tha idea that quarks and leptons can be treated on an equal and that both lepton and baryon number violations are such unified theories, is now an integral part, of GUT their extension.
Grand Unification
611
60
40
a-' 20
0
Figure 3 Running of the three gauge couplings in minimal supersymmetric extension of the standard model. [ref. 51
612
Grand Unification, Supersymmetry and Strings
17.2 S u p e r s y m m e t r y a n d S t r i n g s 17.2.1 Introduction One of the main puzzles in quantum t,heory is how to reconcile General Relativity with quantum mechanics. The usual method of taking the classical Lagrangian and quantizing it fails because of insurmountable difficulties in making sense of the renormalization program, which has been so successful in other quantum field theories. In most situations the domains in which quantum field theories are interesting and the domains in which General Relativity is relevant have no overlap. General Relativity is used when dealing with massive bodies of interest a t large distance scales in astrophysics and cosmology and quantum mechanics is used at, short distance scales. However, there are situations where both theories become relevant. For instance, close to a black hole quantum effects become relevant as evidenced by Hawking radiation. When one begins to probe distances of the order of the Planck scale one expects that quantum gravitational effects will become important. The impasse in the field theoret,ic approach to gravity can be circumvented by using string theory. String theory is a novel program which replaces the plethora of particles that exist by a single string! In this approach the vibrational modes of the string correspond to different particles. Whereas in field theory it seems virtually impossible to include dynamical gravity, in string theory quite the opposite situation prevails: one cannot have string theory without gravity! This is because in the spectrum of string theory there is always a massless spin 2 field, which is naturally identified as the graviton. Another feature of string theory is that it requires supersymmetry. Even though there is no conclusive evidence at, the present, time that supersymmetry is a symmetry of the world, supersymmetry is a favored way of resolving some problems in phenomenology beyond the Standard Model. Issues such as the fine-tuning problem due to a fundamental Higgs are naturally avoided in supersymmet-
613
Supersymmetry and Strings
ric theories since the normally large radiative corrections due to a fundamental scalar Higgs are suppressed due to the presence of its fermionic partner, the Higgsino. Similarly the hierarchy problem can also be resolved in this framework. Supersymmetry is thus seen by many as a positive feature of string theory since the theory requires it and one doesn’t have to introduce it by hand. 17.2.2 Supersymmetry Spac&ime supersymmetry is a symmetry which generalizes ordinary Poincar6 symmetry by augmenting the usual generators with fermionic generators. They satisfy certain commutation relations with the bosonic generators and anti-commutation relations with the remaining fermionic ones:
[Qai,P,] = 0, [Qail
Jpl
{Qai, Q p j }
1
(17.15)
=
i(gpY)!Qpi,
=
- b i j ( ~ p C ) a ~ p p Capzij
+
+ (~sC)aj3z~j.
P, are generators of translations and Jpv are Lorent,z generators. Together they generate the Poincar6 group. The fermionic generators Q are in the Majorana representation and C is the charge conjugation matrix so that: Qai =
CapQP.
(17.16)
The index i runs over the number of supersymmetries i = 1, ..,,N. In the simplest, case N = 1, the other cases are known as extended supersymmetries. The 2 and 2’are so-called cent,ral charges, they are anti-symmetric in the indices i,j and commute with everything. They only exist when one has extended supersymmetry. One of the consequences of supersymmetry is that bosonic particles are naturally paired with fermionic ones so that the number of on-shell degrees of freedom of fermions and bosons are the same. Each minimal pairing consistent with a certain amount, of supersymmetry is called a “multiplet”. For instance, in four dimensions the smallest amount, of supersymmetry has four real fermionic
614
Grand Unification, Supersymmetry and Strings
generators and is referred to as N = 1 supersymmetry. In this case one can have an N = 1 “vector multiplet” which consists of a spin 1 gauge boson along with its supersymmetric partner, a Majorana fermion. The fermions and bosons both have two on-shell degrees of freedom. In addition to the vector multiplet, one can have a “chiral miiltiplet,” consisting of a complex scalar and its partner, a Weyl fermion. Again the degrees of freedom are the same, i.e. two. One can have up to sixteen real supersymmetries (usually refered to as M = 4 siipersymmetry) without introducing anything above spin 1 in four dimensions. Beyond that, one has to include higher spin degrees of freedom. Another useful limit?to remember is that if one restricts the highest, spin of the fields to 2, corresponding to the graviton, the maximum amount of supersymmet,ry is generated by 32 real fermionic generators (often referred to as hr = 8 supergravity). The highest space-time dimension in which a supersymmetric theory can be written down with fields with highest spin equal to 2, is 11 dimensions. This is why eleven dimensional siipergravity plays a distinguished role in supersymmetric physics. When supersymmetry is an exact, symmetry, the bosonic and fermionic partners in a multiplet have the same mass. Clearly, this is not seen in nature. For instance, there is no experimentally observed scalar with the same mass as the electron which would qualify as the electron’s supersymmet,ric partner. Phenomenological models then have to break supersyrnmetry. The mechanism of supersymmetry breaking is not well understood, however, once one assumes that siiperymmetry is broken at some high energy scale, one can incorporate in low energy models the breaking by simply introducing terms which break it. The number of such terms can be restricted to soft-breaking terms which are relevant in the infrared. These terms push the masses of the (as yet,) unobserved supersymmetric partners of the known fields up, to account for their unobserved status while carefully avoiding contradictions with well measured data. Supersymmetry is a vast area of research which deserves
615
Supersymmetry and Strings
and has received book-length accounts* In the next, subsection we will content ourselves with a simple example to illustrate the ideas touched on in our exposition.
Supersymmetric Yang-Mills: A n Example
To illustrate the basic ideas of supersymmetry we analyze a toy model: N = 1 supersymmetric Yang-Mills theory. As mentioned earlier, in a minimally supersymmetric model containing a vector field we need to introduce fermions with as many on-shell degrees of freedom as the vector field. A vector meson in d dimensions has d - 2 physical degrees of freedom, whereas a fermion field with n components has n/2 on-shell degrees of freedom. In four dimensions we need to find a fermion field with 2 on-shell degrees of freedom to match the vector field's physical polarizations. Both Weyl and Majorana fermion have 2 real on-shell degrees of freedom. Consider the following Lagrangian: (17.17) where a sum over repeated indices is implied. a is a group theory index and runs over the generators of the gauge group since all fields transform in the adjoint representation of the gauge group:
The fermionic field @ is taken to be a Majorana field: =
(17.19)
This Lagrangian is invariant under the Poincark group and local gauge transformation, in addition it enjoys a fermionic sym*See for instance, J. Wess and J. Bagger, "Supersymmetry and Supergravity" Princeton University Press (1992).
Grand Unification, Supersymmetry and Strings
616
metry:
(17.20) E is an “infinitesimal” spinor which anti-commutes with fermionic = zgPv as defields and commutes with bosonic fields. And ”/II. fined in Appendix A. This fermionic symmetry combined with the Poincard symmetry is known as N = 1 supersymmetry. We can derive equal-time (anti-)commutation relations for the fields $ and A,. There is a subtlety which needs to be mentioned here. Since the field A0 does not have a conjugate momentum one cannot quantize it in the usual way, more sophisticated methods are called for. In the following we pick the gauge A,-, = 0 and agree to impose the equation of motion of the A0 field (Gauss’ law) by hand on all physical states. In this gauge we can write down the following equal-time commutation relations:
{?434,+*@(Y)}
=
[FO4(”),A 3 4
=
Using these commutation relations and using the Majorana condition, we can write down the generators of supersymmetry in terms of the fields: 1 Qrw= -- d 3 z F ~ v ( y p U y 0 ) ~ $ ~ . (17.22)
4
One can easily verify that these generators generate the above siipersymmetry transformations in the gauge A0 = 0: [EQ, $“] =
E
1
{ Q ,$”} = --F;u~’u~ 4 (17.23)
N
1 super Yang-Mills (SYM) has some properties in common with ordinary QCD. For instance, the one-loop beta function of =
String Theory and Duality
617
this theory is given by:
(17.24) The beta function is negative implying that the theory is asymptotically free just like ordinary QCD. Also like QCD, it, is believed that SYM is confining and develops a mass gap. In addition, SYM has instantons which contribute to correlation functions.
17.3
String Theory and Duality
There are five known string theories, which are called the Type I, Type IIA, Type IIB, Heterotic S0(32), and Heterotic Ea x ES string theories. They are at first sight, very different. For instance, the Type I and the two Heterotic theories have half the sixpersymmetries of the Type I1 theories. Similarly, the Type I and Heterotic theories have non-abelian gauge groups while the others don’t,. One key feature that they do have in common is that they are all formulated in 10 dimensions. In 1995, the groundbreaking work of Hull, Townsend and Witten unified these theories. They argued that, while naively the theories had distinct properties, in many cases they were nonperturbatively the same. Many of these properties can be understood by thinking of these t,heories as limits of a single theory: “M-t heory” . The key concept unifying the string theories is called “duality”. The basic idea is simple. Consider a physical system which has two distinct descriptions A and B, say. A is then said to be dual to B, and vice versa. If the two descript,ions are different,, as they must for duality to be non-trivial, there must be mechanisms by which their apparenta disparity can be overcome. Also, their region of validity must be such that one doesn’t find any obvious contradiction. There are many different dualities. We list a few to illustrate the concept. Strong-weak coupling duality. This is a very powerful type of duality which relates a theory A, say, at strong coupling to an-
618
Grand Unification, Supersymmetry and Strings
other theory B at weak coupling. An example of this duality is provided by the Type I and SO(32) Heterotic theories in 10 dimensions. Their couplings are inversely related. Thus when one of them is strongly coupled the other is weakly coupled. Another example is that of the Type IIB theory which is self-dual under strong-weak duality. This means that, the weakly coupled theory is the same as the strongly coupled theory with some fields interchanged. T-duality. In its most general form T-duality relates string theories on different manifolds to each other. An example is of Type IIA on Rg x S' (where S' is a circle) which is dual to Type IIB on R9 x S1. The radii of the two circles are related by RA = ~ ' / R (a' B is the string tension which is the same as the 10 dimensional Planck length squared). Here we find that, two distinct, string theories on different, manifolds (different, because of their radii) are dual. Similarly, we have that Heterotic string theory on R6 x T4 (T4 is the four dimensional torus) is dual to Type IIA on R6x K3 (K3 is a Ricci flat manifold of complex dimension 2). Perhaps the most amazing dualit,ies involve M-theory. Very little is known about, M-theory and yet it, is a powerful tool in string theory. The defining featlure of M-theory is that, at, low energies it is accurately described by 11 dimensional supergravity. One duality states that M-theory on a circle of radius R is the same as type IIA string theory in 10 dimensions with coupling constant gs = ( R / l p ) 3 / 2 (where Z p is the 11 dimensional Planck length). A surprising consequence of this identification is that strongly coupled type IIA string theory develops a new dimension (since in that limit, R becomes large)! Another, similar, duality states that, M-theory on a line segment, is equivalent, to E8 x Es Heterotk string theory. One of the appeals of duality is that it, allows one to formulate the notion of non-perturbative string theory by changing the description. A key method used in establishing duality is to work with the various supergravities which capture the low-energy dynamics of string theories. The field content of supergravity consists of the massless modes of the string theory in question. For instance, the Type IIA supergravity describes the low-energy dy-
619
String Theory and Duality
namics of Type IIA string theory. It, has a number of massless fields of which the bosonic fields are as follows: q5 g,,
B,, A, Apvp
scalar dilaton graviton anti-symmetric 2-tensor abelian gauge field anti-symmetric 3-tensor
(17.25)
(17.26) The anti-symmetric fields all couple to extended objects known as p-branes. Just as a gauge field couples to a point particle) an antisymmetric (p+ 1)-tensor couples to a p-brane. An important example is B,, which couples to the Type IIA fundamental string. We can compare the above field content to that of 11 dimensional supergravity. The massless bosonic content of 11dimensional supergravity is: G,, graviton CpWp anti-symmetric 3-tensor (17.27)
At first sight it, seems to bare little resemblance to the type IIA field content. Recall) however, that M-theory on R9 x S1is supposed to be equivalent, to Type IIA string theory. When we compactify on S' and take the radius to be small we can ignore the dependence of the fields on the compact coordinate, as is usual when one performs dimensional reduction. From the ten dimensional point of view we can make the following identifications:
(17.28) (17.29)
620
Grand Unification, Supersymmetry and Strings
Thus we see that all the fields are accounted for. The dilaton serves as a coupling constant in type IIA supergravity. The usual string-frame dilaton is related. We see immediately that when the dilaton is large the radius of the circle becomes large and type IIA supergravity becomes a poor approximation for 11 dimensional supergravity. We understand this to mean that, Type IIA is a perturbative theory which is non-perturbatively equivalent, to Mtheory on S'. The spect,rum of p-branes is different, in the two theories, but they too are related as above. We illustrate this identification with a few examples. Type IIA string theory has 0-branes which couple to the gauge field A,, in M-theory they correspond to momentum modes along S'. Since momentum is qiiantized in the S1 direction in integer units of 27r/R, where R is the radius of the compact direction, the number of units is naturally identified with the number of 0-branes. A striking difference is that M-theory contains no strings. It does, however, have a 2-brane (membrane) which when wrapped on the S' appears as a string in 10 dimensions as long as one is justified in ignoring scales smaller than the radius of the compact direction. All string dualities have to satisfy consistency checks of the above kind. Fortunately there are many tests one can perform. Here the importance of a distinguished set, of states known as BPS states are particularly useful. BPS states preserve some fraction of the total space-time supersymmetxy, by virtue of which they are the lowest, mass states in their class and are guaranteed to be stable. Many of their properties can be established exactly, even when the theory is strongly coupled.
17.4
Some Important Results
Many new insights have been gained using duality. Although these areas do not, directly touch on finding phenomenologically viable models, some do demonstrate the ability to study phenomena which generically exist in realistic models. We briefly discuss some of
Conclusions
62 1
these below. In the last few years, using duality, considerable progress has been made in our understanding of gauge theories, particularly supersymmetric gauge theories. Significant, results include the demonstration of confinement and c h i d symmetry breaking in four dimensional gauge theories. String theories have been used to study black holes. One of the most exciting new results concerns the problem of black hole entropy. The Beckenstein-Hawking entropy is a thermodynamic quantity which satisfies a generalized version of the second law of thermodynamics. It has recently been given a statistical mechanical basis by relating it to microscopic states of a black hole. Recently, progress has been made in finding a connectmion between gravity and field theory. One manifestation of this has been a proposal that a quantum mechanics model known as Matrix theory captures the dynamics of M-theory. Many checks have been performed to test the ability of Matrix theory to reproduce supergravity calculations with success. Another approach known as the Maldacena conjecture has led to a radically new connection between conformal field theories and supergravity in Ads backgrounds.
17.5 Conclusions We have given just a flavor of the vast and rapidly growing area of supersymmetry and string theory dualities. The interested reader should consult review articles and books for a thorough introduction to the subject. A good place to start is the recent, book by Polchinski (J. Polchinski, “String Theory” Vols. 1 and 2, Cambridge University Press (1998)).
Grand Unification, Supersymmetry and Strings
622
17.6 Bibliography 1. M.K. Gaillard and L. Maiani “New quarks and leptons” Quarks and leptons cargees 1979, p. 443 (Ed. M. Levy et al.) Plenum Press, New York. 2. P. Langacker, “Grand Unified Theories and Proton Decay”Phys. Rep. 72C, 185 (1981). 3. A. Zee, The unity of forces in the universe, Vol. 1 World Scientific (1982). 4. R.’E. Marshak, Conceptual Foundations of Modern Particle Physics, ’World Scientific (1992). 5 . M.E. Peskin, “Beyond standard model” in proceeding of 1996 European School of High Energy Physics CERN 97-03, Eds. N. Ellis and M. Neubert. 6. J. Ellis, “Beyond Standard Model for Hillwalker” CERN-TH/98329, hepph/9812235. 7. Particle Data Group, The European Physical Journal C3, 1 (1999). 8. J. Wess and J. Bagger, “Supersymmetry and supergravity” Princeton University Press (1992). 9. J. Polchinski, “String Theory” Vols. 1 and 2, Cambridge University Press (1998). 10. S.P. Martin, A supersymmetry primer, Perspective in supersymmetry Ed. G.L. Kane, World Scientific; hep-ph/9709356.
Chapter 18
COSMOLOGY AND PARTICLE PHYSICS 18.1
Cosmological Principle ‘and Expansion of the Universe
On a sufficiently large scale, universe is homogeneous and isotropic. This is called the cosmological principle. A coordinate system in which matter is at rest at any moment is called a co-moving coordinate system. An observer in this coordinate system is called a co-moving observer. Any co-moving observer will see around himself a uniform and isotropic universe. Cosmological principle implies the existence of a universal cosmic time, since all observers see the same sequence of events with which to synchronize their clocks. In particular they all start their clocks with big bang. A homogeneous and isotropic universe is described by the Fkiedmann-Robertson-Walker (F-R-W) metric d s2 = c2dt2- R2 ( t )
1 - kr2
1
+ r2 (de2 + sin28 c@~) .
(18.1)
T , 8,$ are co-moving coordinates and the scale factor R(t) is a scale factor for distances in co-moving coordinates and describes the expansion. k is related to the 3-space curvature. With suitable choice of units for T , Ic has the values +1,0, or -1 corresponding to the closed, flat or open universe respectively. For k = 1, the spatial universe can be regarded as the surface of a sphere of radius R ( t ) in four dimensional Euclidean space. This can be seen as follows.
623
Cosmology and Particle Physics
624
Consider a sphere in four dimensional Euclidean space X;
+ X : + xi + xi = R2.
(18.2a)
The line element is
d12 = dx; + dxi
+ dxi + dzi.
(18.2b)
From Eq. (2a), we get x l d x l + xzdxz
+ x3dx3 + ~ 4 d 2 4= 0.
(18.2~)
The fourth element, dx: can be eliminated in Eq. (2b), wing Eqs. (2a) and (2c),and we obtain dl =
dr2 I-'
+ T" ( d o 2 + sin28 d4') ,
(18.3)
R2
where we have used the spherical polar coordinates X I = r cos qh sin 8, = T sin 4 sin 8, x3 = r cos 8. Putting r' = and then removing the prime, we get z2
It is instructive to use the spherical polar coordinates in four dimensional Euclidean space x1 = RsinXsinOcos4 2 2 = RsinxsinOsingl 2 3 = RsinXcosO ~4 = RCOSX.
(18.5)
Then we get d12 = R2 [dX2
dV
=
+ sin2x
(do2
+ sin20 d4')]
R3 sin2x sin 8 d x d0 d$.
(18.6a) (18.6b)
Cosmological Principle and Expansion of the Universe
625
Now using r = Rsinx, we get back Eq. (3). The radius of the sphere and its volume are given by (18.7)
V = J:/,
JI
R3sin2xsin0 dX d0 d 4
= 27r2R3.
The cosmological principle implies [cf. Eq. (l)]
e = R ( t ) T.
(18.8)
Thus the velocity of expansion is given by
(18.9) where (18.10) is called the Hubble parameter. Let, 11s denote by to the present, time and te the time at which the light was emitted from a distant, galaxy. Correspondingly we denote the detected wavelength by X and emitted (laboratory) wavelength by A, of some electromagnetic spectral line. We define the redshift
(18.11) The redshift is experimentally observed and it clearly shows that, the universe is expanding.
Cosmology and Particle Physics
626
The highest, redshift*so far discovered z = 4.89 so that, the Lyman-alpha line appears in the red part of the spectrum arround 7200 A. This implies that = (1 z ) = 5.89. In the matter dominated universe R t 2 / 3 (see below). This gives [with to M 1.5 x 10" yrs, the present age of the universe) t, 21 14 (1.5 x 10" yrs) 2 lo9 yrs. The existence of these high z-objects implies that, by the time the universe was about, lo9 yrs old, some galaxies (or at least, their inner region) had already been formed. For small time intervals since emission compared to H i ' , Eq. (11) takes the form
%
+
N
(18.12) where 1 is the distance to the source.
18.2 The Standard Model of Cosmology The model is described by two differential equations
A c2R2 8nG R2 + kc2 - -= ---p R2 3 3
+
d (pR3c2) p d ( R 3 )= 0.
(18.13) (18.14)
Here the second term in Eq. (13) is due to the curvat,ure, the third term contains cosmological constant A. Cosmological constant A is very small (1111 < 3 x rn-2) and this term is usually neglected except in the inflationary phase of expansion. G is the Newtonian gravitational constant,. In the units ti = c = 1, 2 - MP (the vePlanck mass) M 1.2 x 10'' GeV. Equation (14) expresses the energy conservation. Here p is the density of the universe and p is the isotropic pressure. Note that Eq. (13) can also be put in the form (18.15a)
The Standard Model of Cosmology
627
Differentiating Eq. (13) and using Eq. (14), one gets
_R --_1 A c R
3
--4nG ( p c 2 + 3 p ) 3 c2
(18.15b)
In addition we need the equation of state. We take this to be that for an ideal gas p = nkBT, (18.16) where n is the particle density and Icg is the Boltzmann constant. Icg = 0.86 x lo-'' MeV/K (K: Kelvin). If we take Icg = 1, then the temperature is measured in MeV. In particular 0.86 MeV= 10l°K. From Eq. (13) (A = 0), we have (18.17) where
3 H2(t) 8n G
(18.18)
is called the critical density. It is convenient, to define the density parameter of the universe P a=-.
(18.19)
Pc
Then from Eq. (17), we get
kc2 = R2 ( t ) H 2 ( t ) (R - 1) = Ri ( t ) H i ( t ) ( 0 0 - 1 ) .
(18.20)
Here the subscript, 0 denotes the present time. It, is clear from Eq. (20) that for 0 > 1, the universe is closed, for R 5 1, the universe is open. We note that for nonrelativistic gas (NR) p = mn 3 p
<< p c2.
(18.21)
Cosmology and Particle Physics
628
Then we say that, the universe is matter dominated. For extreme relativistic gas (ER) p
=
-1p c
2
3 p c2 = 3n,kBT
(18.22)
and we say that universe is radiation dominated. Present, universe is matter dominated i.e. p == 0. Thus from Eq. (14), we have d (pR3c2)= 0
or
3 (18.23) 47f where M is just, the mass of the universe. We define another parameter q , called the deceleration parameter
pR3 = constant
q=
=
-M,
RR --,
(18.24)
RJ
From Eq. (13), using Eq. (23), we get
. .. 2RR
2GM 8.rr R = --G R2 3 *
= --
R pR.
(18.25)
Thus
40
=
1 2 00.
(18.26)
2
Thus for qo > , the universe is closed and for qo 5 , the universe is open. We now discuss the three cases Ic = 0, Ic = 1 arid k = -1. We will now put, c = 1. First we discuss the flat universe (k = 0). From Eqs. (13) and (23), we get
R J R = (2GM)”’.
(18.27)
629
The Standard Model of Cosmology
Integration of Eq. (27) with R(t M 0) M 0 gives 9GM R3 ( t )= -t 2 = 2
3M 4l.r P ( t )'
(18.28)
where the last term in Eq. (28) follows from Eq. (23). Hence we have p-' ( t ) = 67rG
t2 (18.29)
For the closed universe k = 1, we have from Eq. (13) [A = 01 8nG pR2-1, 3
(18.30)
where we have put ( d t = R d q ) : (18.31) Integrating Eq. (30) with the help of Eq. (23)) we get R = MG(1 - C O S ~ ) t = M G ( q -sinq). Similarly for the open universe k
R
=
t
=
=
(18.32)
-1, we get,
MG(c0shq - 1) MG(sinhr1- q ) .
(18.33)
All the three cases are shown in Fig. 1. Finally we note that the age of the universe is essentially determined by the matter dominated universe. The radiation era lasts only for a few minutes. Now using [cf. Eq. (29)] (18.34)
630
Cosmology and Particle Physics
t
Figure 1 Plot of scale factor R ( t ) versus time t for closed open ( k < 0) and flat ( k = 0) universe model.
(k > 0),
we have from Eq. (13) &2
-
87~G Ri po - = - k . 3 R
(18.35)
By using Eq. (20) and Eqs. (18) and (19) for the present time, we obtain
[
& = R o H o 1-no+no-
RO1 R
lj2
.
(18.36)
The integration of Eq. ( 3 6 ) gives the age of the universe tu =
f (Go) Hi1,
(18.37a)
(18.37b) with x = R/Ro and R(t N 0) z 0. The function = 1 and < 1 is respectively given by
f (no)for Ro > 1,
The Standard Model of Cosmology 7.r
for RO 2 2 for Ro = 1 - -3, -
N
N
631
>> 1
(18.38a) (18.3813)
1 +RoInRo, for
0 0
<< 1.
(18.38~)
Thus constraint,s on t, give constraints on f2i1'2 H;'. We close this section by summarizing the essential features of the standard model of cosmology. The universe started with a big bang and has been undergoing expansion ever since. This picture is based on two observed facts: 1. The Hubble expansion. The recession of distant cosmological objects was discovered by Hubble in 1920s. They were found to be moving from us with velocities proportional to their distances u = H l . 2. The observation of black body radiation with temperature TO 2.7 K. This is supposed to be relic of the early universe. The observed isotropy of the background radiation (AT/T N provides the strongest direct support of the cosmological principle. The model is characterized by four parameters: (i) The present value of the Hubble parameter
-
HO= 100 ho km s-l Mpc-l.
(18.39a)
Since Hubble parameter is not very well known, it is written with the ignorance factor ho. The present estimate for ho is 0.4
< ho < 1.
(18.3913)
Note that Mpc : Megaparsec = 3 x lo1' km. Thus HO = 3.33 x
1 0 - l ~ho
s-l =
ho(1 x 10'' yr)-'.
(18.40)
The present age of the universe from Eq. (37a) is given by
t, = f (00) H;'
= f (00)
h i 1 x lolo yrs.
(18.41)
Cosmology and Particle Physics
632
(ii) The present, temperature of the cosmic microwave background radiation (CMBR)
To = 2.728 f 0.002 K. (iii) The average mass density Po = 0
0 Pco
1.88 x 1.05 x
hi gm cmP3 }. hi GeV cmP3
Accurate estimate of the cosmological densit,y parameter ficult. Present experimental data is consistent, with 0.1 5
00
5 2.
(18.42) 0 0
is dif-
(18.43)
Correspondingly Eqs. (41) and (38) give [ ~ ( O O=) 0.9, 0.67 and 1.11 for Ro = 0.1, 1 and 21. (iv) The measurement of deceleration parameter qo = can also give an estimate of Ro, but it is difficult, to measure qo. The present estimates for qo are
if20
Of0.5
t,
N
to
1.5 k 0 . 5 .
(6.5 to 10) x 109h;l yrs.
(18.44)
The best bet, on the age of the universe is (16 f 3) x lo9 yrs. This result, put, constraints on Rohi. 18.3 Thermal Equilibrium
Consider an arbitrary volume V in thermal equilibrium with a heat, bath at temperature T . The particle density n,i (2, particle index) at temperature T is given by
633
Thermal Equilibrium
The energy density is given by
(18.46) where
(18.47) and gi are the number of spin states, q is the momentum of the particle and m iis its mass. The sign is for t,he fermions ( F ) and - sign is for the bosons ( B ) . In particular for i = photon, m = 0, g = 2. In writing Eqs. (45) and (46), we have put, the chemical potential pi = 0. For photon p = 0. Since particles and antiparticles are in equilibrium with photons pi = -p; . If there is no asymmetry between the number of particles and antiparticles, pi = p ; = 0. If the difference between the number of particles and antiparticles is small compared with the number of photons,
+
(18.48) and the chemical potential can be neglected. For the photon gas, we get from Eqs. (45) and (46) 3
(18.49)
3
-
r"15 (L) ( ~ B T )2.7~ nly (ICBT). fic FZ
(18.50)
In Eqs. (49) and (50) 5 ( T ) , T = 3 , 4 is the Riemann zeta function. For a gas of extreme relativistic particles (ER), lc~T>> mic2,qc >>
Cosmology and Particle Physics
634
rnic2,we thus get
The entropy S for the photon gas is given by
R3 4 s = -pr ( T ) . T 3
(18.52)
R3 4 s = -p(T). T 3
(18.53)
For any relativistic gas
Thus for a gas consisting of extreme relativisttic particles (bosons and fermions): (ti = c = 1)
4T)
1
=
2 9 ’ m n,-Y(T)
=
2 g’(T) ( k B T ) 3 7r
1.2
(18.54)
(18.55) (18.56) where
(18.57a) CI
(18.5713)
635
The Radiation Era
are called the “effective” degrees of freedom. We note that entropy per unit volume is given by
1 s -s 2n2 -= - g*(T) ( ~ B T ) ~ . (18.58) ICs R3 k s 45 For non-relativistic gas kBT << mic2,we use the Boltzmann distribution
ni = -?!2n2
(c)1; (--) kBT
3
exp
E
z2 dz
(18.59)
kBT
(18.60) From Eq. (55), we get n,i =
[
___ gi
] (kicT)3 [(g)3’2 -
e-m,c2/kBT
(243/2
pi = ni mi.
18.4
1
(18.61) (18.62)
The Radiation Era
For extreme relativistic gas, p =
p c2, we get, from Eq. (14)
R 3d PE + 4pR2 = 0.
(18.63)
Thus, we have p = A2R-4
Hence
-PN.R NR-+O PE.R
(A: constant,).
(18.64)
R+O.
Therefore, we have the .important, result,. Early universe is dominated by extreme relativistic particles, i.e. the universe is radiation
Cosmology and Particle Physics
636
dominated in early stages. Since p --t $ for t,he early universe we can neglect the second and third terms 011 the left-hand side of Eq. (13) as compared with the first, term. Thus we get
(18.65) Therefore, the expansion rate is given by
(18.66) Now using Eq. ( 5 5 ) , we get
Also we have
RR=A
(18.68)
Hence we get
(-)MeV
= 2.42 g;1/2
Thus as t
s.
(18.69)
+0
Ht
M
0.5.
(18.70)
We consider two examples: (i) For g* = gr = 2, we get
t
M
(-)
1.7 MeV kBT
s.
(18.71)
637
The Radiation Era
'Thus for lc~T= m,c2 = 0.51 MeV, t = 6.5 s and H 0.08 s-'. (ii) For m, > lc~T > me,
7
M
+
g* = 97 + g(ge 39,) = 2 + -7( 4 + 6 ) = - 43
8
4
(18.72a)
and for m, > l c ~ T> mP, 9* = 2 +
s7
(ge
+g, +39Y)
57 - 4
(18.72b)
Here we have taken the number of neutrinos N, = 3. Now we get from Eqs. (67) and (69) at lcsT = 1 MeV
(g) 2
H
N"
0.67
sU1 M 0.67 s-l
(18.73a)
and
t x 0.74
(-)MeV
s x 0.74 s.
(18.73b)
Now Eq. (69) gives the time evolution of the universe in radiation era. From Eq. (66), we have the important, result, that H 0: fi i.e. the higher the energy density in the early universe, the faster will be the expansion rate. As we have seen, the radiation density falls off as and The the energy density in nonrelativistic matter falls of as universe eventually becomes matter dominated. At, t = teg,matter density becomes equal to radiation density i.e. Pm = PT,
(18.74)
Cosmology and Particle Physics
638
where from Eqs. (34) we get (18.75) and from Eqs. (50), (51) and (64) we get
(18.76)
(2)3
where we have used = A,(cf. Eq. (113)) and Tyo.= To. Hence from Eqs. (75) and (76), we obtain for To = 2.728 K, 1
+
Zeq
= =
RO = Ro pco (2.27 x lo6 MeV-'cm3) Re, 2.4 x lo4 Clo hi. (18.77)
From Eqs. (75)-(77), we get kBTeq =
RO = 5.6 x (ICnTo)-
Ro hi MeV
(18.78)
Re,
and from Eqs. (69) and (78), we obtain
teqM 3.0 x 10'' (Ro hi)-' s.
(18.79)
In the dense early universe, the radiation would have been held in thermal equilibrium with matter and would have scattered repeatedly off free electrons. But, when the expansion had cooled the matter below 3000 K ( ~ BNT 0.26 eV), so that, from Eq. (69)
The Radiation Era
639
+ *: T ,
with g* = 2 (6) = t 11 1.3 x 1013 sec N 4 x lo5 yrs, the primordial plasma would have recombined to atoms, the universe thereafter becoming transparent to light. The experimentally detected microwave photons are therefore direct messengers from an era when the universe had an age of about 4 x lo5 yrs. But photons are still around-they fill the universe and no where else to go. The thermal radiation last scattered at this epoch is now detected as the cosmic background radiation. This epic defines a “surface” known as “surface of last scattering”. Some important dates in the evolution of the universe are given in Table 1. These are only estimates. We end this section by writing some useful numbers. From Eqs. (49) and (50), using the present, temperature TO= 2.728 K, we get 2.4 M 411 cm-3 ny0M (18.80) 7r2
(y)’
pro M 2.6 x 10-l’ GeV ~ m - ~ .
(18.81)
Thus n7 at temperature T is given by n7 M 411
(-)T
3~ m - ~ .
2.728
(18.82)
In addition we write down the following estimates. There nucleons in a typical star. There are about, 10l1 are about galaxies in the universe, each galaxy has about 10l1 stars. Thus there are about lo7’ baryons in the universe. This is to be compared with lo8’ photons within the part of the universe we can observe; this number is obtained by thermodynamical argument,s. Thus number of baryons/number of photons M lo-’’. The present, cm. Further the baryon numsize of the observable universe is ~m-~. lo’’ ber density n b is given by n.b N
(
(1029
.>
N
Another quantity of interest is baryon density in the universe. First we note from Eq. (23), that it scales as K 3 . It is
Cosmology and Particle Physics
640
Table 18.1 Cosmic History (some critical phases)
Era
Planck
Age (in seconds)
Temperatm e
K
Remarks
0
Vaciiiim to matter t,ransition
10-44
All forces unify
GUT transition,
GUT
Strong and electxoweak forces unify, Baryon number creation
10-36
w*, zo:
Electroweak
10-’O
Quark
10-5
3 x 10l2
Hadronization
Lepton
8 x lop5 8 x lop3 6
1.2 x 10l2 1.2 x l o l l 1o1O 6 x lo9
,uk annihilation up’s decouple u,’sdecouple e+e- annihilation
Particle
60 - 80
[1.3 - 0.81 x lo9
Nucleosynthesis
Photon
6 x 10l2 to 1014
4 x 10l2 to 103
Radiation era ends Plasma to atom;
2 x 1017
2.7
Present
10l8
Salam-Weinberg transition
0.7
The Radiation Era
641
convenient to define the baryon density in terms of parameter 71 viz. (18.83) where n B = n b
- "6.
The baryon number density is given by
(18.84) where p~ is the baryon energy density and OB = p B / p c . Now using [cf. Eq. (42)j pco x 1.1 x hi GeV ~ r n - ~ (18.85) and taking
mB =
1 GeV, we obtain
This relation is sometimes written as f l B o h i = 3.78 x lo7 q
(18.87b)
and PBO
= q n,
(1 GeV) = 17 (411) (1 GeV) cmW3
= 7.0 x 10-22q gm ~ r n - ~ .
(18.87~)
Big Bang nucleosynthesis limit q to [see Sec. 71
2.4 x lo-''
5 q 5 4.2 x 10-l'.
(18.88)
Cosmology and Particle Physics
642
18.5 Freeze Out
At high temperatures ( k B T
>> m ) )thermodynamic equilibrium is
maintained through the processes of decays, inverse decays and scatterings. As the universe cools and expands, the reaction rates will fail to keep up with the expansion rate and there will come a time when equilibrium will no longer be maintained. At, various stages then, depending on masses and interaction strengths, different, particles will decouple with a “freeze oiiV surviving abundance. We now determine conditions under which the statistical equilibrium is established. From dimensional analysis, the reaction rate for a typical process can be written as follows. For the decay of a X-particle, the decay rate is given by
2
is the measure of where rnx is the mass of the X-particle, ax = coupling strength of X-particle to the decay products, and gd are number of spin states for the decay channels. Note that
(18.90) The reaction rate for the scattering processes is given by
I‘= (cv) [number of target, particles per unit, volume]. (18.91) For a weak scattering process
.(
4 = gtv
(18.92)
Freeze Out
643
Since the number of target, particles per unit volume n, we can write the reaction rate for a weak process
N
(~cBT)~,
(18.93)
For lc~T<< mw,we get
(18.94) The condition for thermal equilibrium is
rrH
(18.95)
i.e. the reaction rate I' must, be greater than the expansion rate t80 maintain the thermodynamic equilibrium. We now consider a specific example. At, about, a temperature of 10 MeV, the universe is made up of neutrons, protons, v's, V's , e* and 7's in thermodynamic equilibrium. At about a few MeV, the neutrinos decouple. To see this consider the processes
For these processes (c v) Thus the reaction rate
=
G;/n s and (2/3.rr)G$ s respectively.
2 -G; (~BT)~. 3.n Now using Eq. (73), we find from Eq. (96) [GF GeV-2]
(18.96) =
1.166 x
(g) 2
0.7
2 3.rr
= -Gg = 0.04
s-'
(1 GeV)410-12
(-)kBT MeV
5
s-'.
(18.97)
Cosmology and Particle Physics
644
Thus the decoupling temperature for neutrinos is given by
kBTD
=
2.6 MeV.
(18.98)
Hence for kBT < 2.6 MeV, neutrinos are decoupled. The neutrinos are extreme relativistic partsides. For (ER) particles
NFR hq = n,:," V
0: T 3
R3.
(18.99)
Now the entropy for ER gas is given by [cf.
S=
R3 4
--
T 3
p(T)
(18.100)
But p ( T ) 0; therefore, S ( l / R T ) . Thus for the entropy t o remain constant T o( R-'. Hence from Eq. (99), we have the important, result: In equilibrium ER particles are conserved. This can also be seen as follows: As we have discussed in the beginning of this section, at high temperatures all interacting species i, j , 1, m are in thermodynamic equilibrium through the react,ions of the type N
ij-lm.
As the reaction rate r < H (the expansion rate), the species involved decouple and their abundance is frozen out. Consider an arbitrary volume V and let Ni be the number of particles of type i in this volume. Thus for the reaction i j t i 1 m, we have (18.101) where
Freeze Out
645
Now Ni = niV, therefore
d Ni -dt
d
ni
- v - d t + ni
d V d ni d t 0; [R3-d t + 3 ni R 2d te ] . (18.104)
Hence we have from Eq. (103) d ni R - - -3 ni dt R
+ (V
Olm+ij)
nm nl
- (71 oij+,lm)
n,j nIi. (18.105)
The principle of detailed balance gives (18.106) Thus from Eqs. (103) and (105), we have
and b
(18.108) First, we note that n.7 etc. are given by Eq. (45). For d NF -0 dt
-I
(18.lO9)
i.e. for the conservation of N Y , we must have d n? = -3 niEq R -. dt R
(18.110)
From this equation, we get
niEq
0;
-
R3'
(18.111)
Cosmology and Particle Physics
646
and from Eq. (107), we get nj)Eq = (721 nm)Eq
i.(
i.e. in equilibrium extreme relativistic particles are conserved. The condition (111) is always satisfied for the ext,reme relativistic particle [cf. Eq. (64)]. Weakly interacting particles may decouple when they are ER, massless particles are always ER. For massive particles whose interactions are sufficiently strong to be capable of maintaining equilibrium when Icg T < m : [cf. Eq. (61)].
NE”q”.= n,:: 18.6
V
0;
1
( m kB T)3’2exp
(18.112)
Limit on Neutrino Mass
We now use the result t,hat, neutrinos decouple at a temperature of a few MeV (i.e. they go out of equilibrium before e-e+ annihilat,ion heated up the photon background radiation). Thw Tvowill be less than TTo.Using Eq. (51), we get
(18.113) Now using Eq. (53) or is given by
(loo), the entropy before e-e+ R3
4
s = -3
b e -
+ Pe+ + P-l) Tbefore -1
annihilation (18.114)
and the entropy after e-e+ annihilation is given by
s = -4p 3
-,
R3
(18.115)
’Tafter
Thus we have from Eqs. (114), (115) and (50)
(;
x 2
+ -87 x 2 + 2,
T:efore= 2
(18.116)
Limit on Neutrino Mass
Noting that
Tbefore
647
= T,o and
=
($)3
, we get
= T,o
?'after
11. 4
(18.117)
Hence we have from Eq. (113) %o - -_ -3 n,yj 11
(18.118)
and [cf. Eq. (82)] n,~ =3 (411)
~ 1 2 1 ~ ~ . (18.119) 11 The present neutrinos density in GeV cm-3 can be written as
x
GeV ~ m - ~ . (18.120)
Using Eq. (119), we get
Now p,o must be less than the average density of the universe po. Thus [cf. Eq. (42)]: p,o
< po
Ro hi GeV ~ r n - ~ .
x 1.1 x
(18.122)
Hence we have from Eq. (121)
x m , i < 100 Ro hi eV.
(18.123)
2
Here the sum runs over all neutrino species with mu < 1 MeV. Now if the age of the universe t , 2 13 x lo9 yrs, then, Eqs. (37) and (38) imply that Ro hi 5 0.45 for ho 2 0.4, while t, 2 10 x lo9 yrs implies Ro hi 5 1 for ho 2 0.4. The above constraints on Ro hi give respectively < 45 e~ (18.124a)
Emui i
Cosmology and Particle Physics
648
or
C m v i < 100 eV.
(18.124b)
a
There is a wide consensus among the astrophysicists that at least 90% of the mass in the universe does not shine (i.e. not, visible by optical or radio means). It is only detected through its gravitational interaction. Now from Eq. (87c), p ~ M o 7.0 7 x
gm/cm3
(18.125)
where as discussed in the next section [cf. Eq. (135)J 7 5 4 x 10-l' givjng pB0
5 2.8 x
gm/cm 3
(18.126)
to be compared with the critical density pCo M 1.88 x gm/cm3. Furthermore from Eqs. (87b) and (88)
hi
or, since 0.4 5 ho 5 1,
Since Ro > 0.1, the dark matter is mostly of nonbaryonic origin and the universe is not closed by baryons. It! must, certainly have contributed to the formation of galaxies. Light. relic neutrinos are typical candidates for hot dark matter because their velocity was relativistic at the t,ime of decoupling. Eqs. (42) and (121)
(18.128)
xi
Unfortunately there is no direct particle physics evidence an m,. However, if f l F D M N 0.2 and hi N 0.30, then mvi N 5 - 6 eV.
xi
649
Primordial Nucleosynthesis
18.7 Primordial Nucleosynthesis
--
At temperatures 2 1 MeV, the weak reactions such as ~,+p e-+p
e++n v,+n
(18.129)
are still fast compared with the expansion rate of the universe to maintain thermodynamic equilibrium between p and n. The abundance ratio at equilibrium is given by (18.130) Using Am = (m, - mp) = 1.3 MeV and k B T = kBTD = 1 MeV, we find n / p = 0.27. The decoupling temperature TD is est,imat,ed as follows. The rough estimate for the reaction rate in Eq. (129) is given by Eq. (94). A more accurate calculation gives
I’
=
77r
z G $ (1+3g;)(k~T)’ 4.22 G; (ICgT)’ = 0.8
The decoupling temperature is given by where we have taken N , = 31
(s)’ s-’.
r =H
(18.131)
viz. [cf. Eq. (73),
(g)‘(g)
(18.132)
k B T = kBTD N 1 MeV.
(18.133)
2
0.8
= 0.7
Thus
As the temperature cools past the decoupling temperatiire kBT~ x 1 MeV, it is no longer possible to maintain the thermal equilibrium. The ratio n / p thereafter is frozen out and is approximately constant (it decreases slowly due to weak decay of neutron). The freeze out, n / p ratio is given by ( 18.134)
Cosmology and Particle Physics
650
where we have used the Q-value Q = (ma- mp)+me = 1.8 MeV. For T > Ts, the deuteron formed is knocked out by photo dissociation y D p n,, since the binding energy AB for the deuteron is only 2.2 MeV. The formation of deuteron actiially st,arts after k B T 9 = 0.1 MeV; TS is called nucleosynthesis temperature. The estimate that, kBTs M 0.1 MeV can be obtained as follows:
+
-+
+
(18.135)
Thus -
(18.136)
Using A B = 2.2 MeV and 7 = lo-”, we find k ~ T = s 0.1 MeV. For T > Ts, photodissociation is so rapid that deuteron abundance is negligibly small and this provides a bottleneck to further nucleosynthesis. The deuteron “bottleneck” thus delay nucleosynthesis till k B T 5 0.1 MeV. But once the bottleneck is passed, nucleosynthesis proceeds rapidly and essentially all neut,rons are incorporated into 4He : n+p
D+D 3
~
3 H + 4He
-+
D+y 3
-++
~ + p , ~ ~ , + n 4 ~ ~, + n ,
-+
’Li
-+
It is clear from the above reactions that 4He abundance is given by 2 ( n / p ) - 0.32 Y= - - = 0.27. (18.137a) l + n / p 1.16 The ratio Y changes from T, to Ts due t,o the neutron decay n + p e- V , . During this time n / p changes from 0.16 to 0.14. Thus at, T = Ts, 0.28 y r = 0.25. (18.137b) 1.14
+ +
651
Primordial Nucleosynthesis
We conclude that n / p ratio or Y depends on three parameters: (i) decoupling temperature TB, which in turn depends on the number of light particles, e.g. number of neutrino flavors Nu.
(ii) neutron decay in between To and T , i.e. on the decay rate of neutron or neutron half-life r1p.
In fact, Y is most sensit,ive function of r / H . Now TD depends on the expansion rate H ; the expansion rate depends upon the effective degrees of freedom g*, the higher the g*, t.he faster the expansion rate. This implies higher To and hence higher n / p freeze out abundance. Thus the higher the g*, the higher will be Y. But g* = 2 i ( 4 + 2Nu) , where Nu are the number of neutrino species. For Nu = 3, g* = and we obtained TO M 1 MeV and Y x 0.25. The observed primordial abundance of 4He gives Y = 0.234 f 0.002 (&0,005). The half life for neutron decay, the parameter needed in the above analysis, is r1p = 887 f 2 s . Taking Nv = 3 as given by LEP data [cf. Sec. 131: Nv = 2.999 It 0.016, one can use the observed primordial abundances of D , 4He and 7Li, to get, a limit on 7 :
+
2.4 x lo-''
5 7 5 4.2 x lo-''.
(18.138)
As already remarked this value of 7 , implies [cf. Eq. (127b)l 0.009 5 f2;2B' 5 0.1.
(18.139)
If Ro > 0.1, then as remarked in the previous section, some other non-baryonic form of matter must account for the difference between !& and f 2 ~ 0. There are some dynamical models which suggest Ro = 1, this requires a large amount of non-baryonic matter.
652
18.8
Cosmology and Particle Physics
Baryon Asymmetry of the Universe: Baryogenesis
There is no evidence for the existence of antibaryons in the universe. The baryons to photons ratio q = nB/n7 N 3 x 10-l'. The asymmetry between baryons and antibaryons can be explained as follows: The universe started with a complete matter-antimatter symmetry in a standard big bang picture. In the subsequent evolution of the universe, a net baryon number was generated. This is possible if the following three conditions are satisfied: (i) There exists a baryon number violating interaction.
(ii) There exist C and CP violation to introduce the asymmetry between particle and antiparticle processes. (iii) Departure from thermal equilibrium of X-particles which mediate the baryon number violating interactions. The condition (iii) is necessary because if the baryon - violating interactions were always in equilibrium, the number of particles and antiparticles would be given by e - m / k B T and e - m / k B T and thus would be equal since f i = m by CPT theorem. The condition (iii) is supplied by the expansion of the universe. The condition (i) is supplied by the X-particles (vector and scalar bosons) predicted by grand unified models. At T = TO (the decoupling temperature i.e. the temperature at which X-particles go out of equilibrium), the number density of X-particles is given by [cf. Eqs. (49) and
(WI: (18.140) where gx is the total number of X (and entropy density at TDis given by
X )spin states. Now the
(18.141)
Baryon Asymmetry of the Universe: Baryogenesis
653
where g* is the effective number of degrees of freedom. The number of baryons at T O are given by n,B
= n,xD
AB.
(18.142)
But
(F)
AB
0.28
=
(18.143)
Now g+ is over 100 in a typical GUT. [ In SU(5): y, W*, Zo,8G's, 34 Higgs, 6 quarks, 3 leptons, 3 neutrinos, 12 X's. Thus g* = (24 x 2) 34 i ( 1 8 x 4 3 x 4 3 x 2) = 160.8.1 We, therefore, expect g x / g * x to 10-l. Thus we have
+ +
kg
(y)
M
+
+
0.28~
-
lo-')
AB
M
3x
-
AB.
D
But ( n ~ / s =) (~~ I C ~
B / s
( y )=
(18.144) )where ~ , 0 denotes the present time. Thus
3 x (10-3
- 10-2)
AB.
(18.145)
0
Now [cf. Eqs. (58) and (54)]
Hence from Eq. (145), we get,
x 21 x (10-3 to
10-2)
AB
M
2x
to 10-l) A B .
(18.147)
Cosmology and Particle Physics
654
The X-particles can generate AB, by the processes of the following type
X X X X
B1 = 1/3
-+
ql : r
-+
QQ: 1- r
-+
qZ: F
+
4 4 : 1- F
Ba 8 1
=
-213
-1/3 Ba = 213.
=
The mean baryon number per decay
(18.148) Thus
1 2 1 = -( 2
= - [r
( B -~ B2)
+r
(Bl - B ~ +) ( B+~B ~ ) ] (18.149)
T - F ) .
F’rom Eqs. (149) and (147), we see that, we can explain the baryon number generation if r # F , i.e. X-interactions violate C and CP. Also we require A B in order to explain the present, baryon number q = nB/ny z 10-l’. Let us now obtain an estimate for To. If Icg To > mx,the thermal equilibrium can be maintained by inverse decays. Thus the condition for departure from equilibrium is [cf. Eqs. (67) and (9011: N
Now using gd
%
12 x 2 kg
=
To
24 and g* QX
= 160, we get
(4.0) 10l8 GeV.
(18.151)
Inflation
Using ax
655 M
1/40 [SU(5) value], we get kg TDM 1017 GeV.
(18.152)
Thus if X-boson are vector bosons, k g TD> mass of vector bosons of SU(5) and therefore vector bosons of SU(5) cannot give rise to baryon asymmetry. lop4, and k g TD 1013 However, for Higgs scalar Q X GeV. If kgTD < the mass of Higgs scalars, inverse decays are not, energetically allowed and baryon asymmetry may arise as the scalar bosons go out of thermal equilibrium at kg TO 1013 GeV. However, in SU(5), the scalar bosons give A B but we require N
N
N
N
AB
N
18.9 Inflation There are several problems in the standard model of cosmology. We now discuss two of these problems and how to resolve them in an inflationary universe. 18.9.1 Horizon problem Horizon of the universe r ~ ( t(called ) the particle horizon) at, time t is defined as the size which can be causally related during the evolution of the universe. Since signals cannot, travel with speed greater than c , r&) = ct r H ( t 0 ) = cto M
cm, [to
N
4.5 x lo1' s]
(18.153)
This gives the maximum size of the observable universe. We call the present size of the universe ro(t0). Let us extrapolate it backward in time: (18.154) Thus (18.155)
Cosmology and Particle Physics
656
Let us take t = t, = 1.3 x 1013 sec (4 x lo5 yrs), which corresponds to the epic of the “last scattering surface”. This is the time when the universe just, started to be matter dominated. Thus [cf. Eqs. (69) and (29)]
R(t)
{
Iv
t1I2 t2I3
t < t, t > t,.
(18.156)
Hence, rewriting Eq. (155),
(18.157) Then using to M 4.5 x 1017 s (1.5 x lolo yrs), we have at the last, scattering surface ro ( t d 33
m-
while
O‘ r H (tGut)
loz6,
tcUl M
S.
(18.158)
The uniformity of the temperature of the background microwave radiation _< provides a strong evidence that universe is isotropic t,o a high degree of precision. But, the present size of the universe extrapolated backward in time to t, is 33 times the particle horizon. Therefore, there would not have time for transport processes to equalize the temperature over the last, scattering surface. Thus it is difficult to explain the high degree of isotropy in the present universe.
(?
Inflation
657
18.9.2 F l a t n e s s problem Why is the universe near the critical density? Stated in other words why the curvature term does not, dominate at a certain R(t). To see this problem, we note [cf. Eqs. (20) and (17)]
( 18.159)
(18.160) Using Eq. (156), (18.161) Thus we get
(18.162) Taking t , 1.3x 1013 s and t o and 0 0 = 21 N
(t,) while for
tplan& N
N
N
4.5 x 1017s , we have [for RO= 0.1
1 + 1 0 - ~(0,
-
1) M I
10-~
(18.163)
s (tplanck) =
1
(18.164)
Such a terrible “fine tuning” looks unnatural. The natural solution is either p = pc, R = 1 (for a reason to be discovered) for the whole history of the universe or some nonstandard mechanism intervened to derive po pCo, Ro + 1. ---f
Cosmology and Particle Physics
658
18.9.3 Inflationary universe The basic idea of this scenario is that there was an epoch when the vacuum energy density dominated the energy density of the universe. Thus we write
-
(18.165)
&
The radiation era density pT but, pv is constant, independent of R. Suppose pv >> pT in the early universe. Thus from Eq. (62), we have R =H= = const,. (18.166) R 3 Thus pv acts like an effective cosmological constant,. We get N
where we have put,
(18.168) The exponential increase of R ( t )with t is called the inflation. What can cause this inflation? This scenario may happen in the spontaneously broken grand unified theories (GUT). Consider the phase transition for the symmetric phase [(+) = 01 T >> T, to the broken phase [($) # 0 ] T < T, . If this phase transition is of first, order, then it is accompanied by latent heat. Note that, T, denotes the crit,ical temperature and 4 is the Higgs scalar responsible for spontaneous symmetry breaking. For T >> T,, ($) = 0 is the local and global minimum. At T = 0, (4) = Mx is the local and global minimum. For T = T,, both (4) = 0 and (4) M x are minima. Below T < T,, (4) = 0 is a local minimum (false vacuum) (see Fig. 2). If the universe is trapped in t,his false vaciiiim, then it is in a stage of supercooling because to go over to true vacuum it has to N
659
Inflation
cross a potential barrier (see Fig. 2), either by thermal fluctuations or by quantum mechanical tunneling. If V is sufficiently flat,, the time required for 4 to transverse the flat region can be long compared to the expansion time scale tGut, say r@M 65 tGut. During this slow growth phase pv = V ( $= 0) dominates over pr and we get R ( t )= etltG"t, (18.169) where (18.170)
-
Here we have put, pv M i T:, Mx M 1015 GeV and G GeV-2. Thus for t = 65 tcut = r , we have N
4M MP
Hence we have =
1.75 x lo-' e e-65 (1053)1126,
N
lo-'
r H (tGut)
< 1.
-
(18.172)
Note that without the inflation TO (tcut) Thus . ./ r H (tcut) the problem of causal disconnection.is solved. 'We note that tGtt M 10'lGeV cm. But it exponentially grows to e10010-25 cm after t = 100 tGut M s. N
N
Cosmology and Particle Physics
660
The flatness problem is also solved in the inflationary universe scenario. Now [cf. Eq. (17)] (18.173)
Also [ R ( t ) ] - 2 e-2t’tcut --t 0 and p = pv (constant) for inflationary epoch, we get from Eq. (173) N
R ( t )M 1.
(18.174)
Hence we see that R is driven to 1 in inflationary scenario. The latent heat of the phase transition is used to reheat the universe to T M 1014GeV,thus making baryon synthesis and creation of baryon number possible. We have not discussed the monopole problem at all. We have only sketched the inflationary universe scenario. A more detailed discussion of inflation is beyond the scope of this book. Figure 3 gives a very rough sketch of the inflationary universe.
661
Inflation
Figure 2 Behavior of the potential term VT($)for T and T >> T,.
Inflation
< T,,T
= T,
I\
Figure 3 Scale factor R(t) versus t, showing the inflationary phase.
Cosmology and Particle Physics
662
18.10
Bibliography
1. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (4th Edition) and Statistical Mechanics (3rd Edition), Part I, Pergarnon Press 1985. 2. P. J. E. Peebles, Physical Cosmology, Princeton University Press, Princeton, N. J. (1971). 3. D. W. Sciama, Modern Cosmology, Cambridge University Press, Cambridge (1972). 4. S. Weinberg, Gravitation and Cosmology (Wiley: NY, 1972). 5. G . Steigrnan, Ann. Rev. Nucl. Part. Sci. 29, 313 (1979). 6. F. Wilczek, Erice Lecture on Cosmology, Proc. 1981 Int. Sch. of Subnucl. Physics, “Ettore Majorana”. 7. A. Zee, Unity of Forces in the Universe Vol. I1 (World Scientific, Singapore 1982). A collection of original papers relevant to this chapter can be found in this book. 8. M. S. Turner, Cosmology and Particle Physics, Lectures at the NATO Advanced Study Inst. Edited by T . Ferbal, Plenum Press (1985). 9. D. Denegri, The Number of Neutrino Species CERN-EP/89-72. Rev. Mod. Phys. 10. A. D. Linde, Particle Physics and Cosmology, in Proc. XXIV Int. Conf. on High Energy Physics (Editors R. Kotthaus and J. H. Khn) , Spinger-Verlag, Heidelberg, Germany (1989). 11. A.H. Guth, The Inflationary Universe, Addison-Wesley, Mass. (1997). 12. R. Kolb and M. Turner, The Early Universe, Addison and Wesley, California, 1990. 13. Particle Data Group; Eur. J. Phy. C 3, 1-794 (1998).
Appendix A QUANTUM FIELD THEORY [A SUMMARY] A.l
Spin 0 Field
Spin zero particle of mass rn is described by a field +(z) which in the absence of interactions, satisfies the Klein-Gordon equation.
In quantum mechanics, 4 ( x ) is regarded as a c-number. In quantum field theory, +(z) is a field operator which can create and annihilate the field quantum. The Fourier decomposition of
4(x) is
(A.2b) where $t(x) is hermitian conjugate of 4(z) and k . x = Icoz~k.x, ko = d m > 0. In Eq. (2), u ( k ) and b ( k ) are interpreted as follows:
663
Quantum Field Theory [A Summary]
664
at (k) : creation operator for the particle (spin 0 and mass m ) annihilation operator for the particle a (k) : (spin 0 and mass m ) bt (k) : creation operator for the antiparticle (spin 0 and mass m ) b (k) : annihilation operator for the antiparticle (spin 0 and mass m).
a (k) and b (k) satisfy the following commutation relations
S3 (k- k’)
(A.3a)
[ b ( k ) ) bt (k’)] = S3 (k- k’)
(A.3b)
[ ~ ( k ) at , (k’)]
I)’+
[a@),
=
=
[ a @ ) ,b+(k’)]= 0.
(A.3c)
If 10)denotes the vacuum state then one particle state of 4-momentum k is given by
lk) = a+(k)10) .
(A.4)
Define
N+ (k) N- (k)
= =
at@) u(k) bt(k) b ( k ) .
(A.5a) (A.5b)
It follows from the commutation relations (3) that N+ (k) and N- (k) have the eigenvalues 0, 1, 2, and are known as number operators for the particles and antiparticles. Then
-
n
=
c N+ (k) c N- (k)
= Total
number of particles
(A.6a)
k
A =
= Total number of antiparticles. (A.6b)
k
Spin 1/2 Particle
665
It may also be noted that for free fields
(A.7a)
@o)
=
i
>o
Ico Ico
+1 -1
(A.7c)
We note that
viz. the spacelike distances. Then from (7a), it follows that the commutator is zero for space-like separation. This is the statement of the micro causality. Also
and from Eq. (8), we get
A (0, X - y ) = O .
(A.9b)
A.2 Spin 1/2 Particle Spin 1/2 particle of mass m is described by a field !I!(x),which is the absence of interactions, satisfies the Dirac equation [a, = =
&
(&V,] (iy”a, - m ) !I!(x)= 0.
(A.lOa)
The adjoint of !I!(z), $ (x)= !I!t (x) yo satisfies the equation
+ >
G ( x )(-27, a,
-m
7’”are Dirac matrices. We choose y” :
= 0.
(A.1Ob)
Quantum Field Theory [A Summary]
666
(A.ll) -f ’s satisfy the anticommutation relation
(A.12)
Matrices 1
I
I
y5 is also hermitian
Y5 t -7
Components 1 4
5
I
(A.13)
y5 anticommutes with yP viz.
Y5 YP -- -YP Y5*
(A.14)
In Pauli representation, yP ’s can be written as
(A.15a)
Yo =
r’
=
(; iil)
(-9;
The Fourier decomposition of
(A.15b)
=yo
Z)’
(x)is
75’7
5
(A.15~)
667
Spin 1/2 Particle
(A.16b) where E = t,
and u and
21
yo,
jj = t.
yo
(A.17)
satisfy the equations
( 7 . p - m, uT ( p ) (Yap+ m, vT ( p ) G ( p )(74- m) %(p)(y.p+m)
=
(A. 18a)
=
(A.18b) (A.18~) (A. 18d)
0 = 0 =
u ( p ) and b(p) are interpreted as follows:
creation operator of the particle wit,h momentum p and spin component T annihilation operator of the particle with momentum p and spin component r creation operator of the antiparticle with momentum p and spin component r annihilation operator of the particle with momentum p and spin component r. The operators a and b satisfy the anticommutation relation
Quantum Field Theory [A Summary]
668
and all other anticommutation relations give zero. Define number operators:
N,(+)( p ) = N,(-) ( p ) =
bL
(p) (p)
(PI bT
( p )'
(A.20a) (A.20b)
Then from the anticommutation relations (19), we have [A$*) ( p ) I 2 = N,(*) ( p ) .
(A.21)
Thus, we see that A$*) ( p ) have eigenvaliie 0 or 1. This means that each state is either empty or has a single particle of definite spin and momentum. Thus the anticommutation relations lead to description of a system of particles which obey the Pauli exclusion principle or in other words obey the Fermi-Dirac statistics. The spinors u and u satisfy the following orthogonality relations:
They also satisfy the completeness relations (A.23) where a and
p are spinor indices; a , p = 1 , 2 , 3 , 4 .
(A.24a)
(A.24b)
Spin 1/2 Particle
669
A+ ( p ) and A- ( p ) are the projection operators for particles and antiparticles respectively. One also writes y, p , = y p = $. Using the Pauli representation of y-matrices, we can write
u,( p ) = R
(A.25a)
where
(A.2 5b) (A.25~) =
ar( p ) is given by
( ;)
and
d2)=
-
r)t
u, ( p ) = ~ ( ‘ 1 Rt ~ yo = W (
where
R=
1
JGjZxj((Po + m) 1,
-0
( ).
(A.25d)
-
R,
(A.26a)
‘P).
(A.26b)
w, ( p ) is given by
( p ) = -iY2 u:: (P) Finally, we note that for free fields [pz= ypa,] 21,
p a (4
1
-
%
(igz+ m ) ,
= i = -2
s, p
(34
+
A (Z- d)
(x - d),
where
S
(X - d) =
(A.27)
(-igz - m) A (Z- d).
(A.28a)
Quantum Field Theory [A Summary]
670 -
[@ (x), @ (z’)]
A.3
.
= yo b3 (x - x’) +so=*;
(A.28~)
Trace of y-Matrices
We note that, y-matrices are traceless
T r y’l = 0, T r y5 = 0.
p = 0,1,2,3
(A.29)
Now
Tr (y’l 7”) = Tr (7”7’1).
(A.30)
Therefore, from Eq. (12), we have
T r (y’l 7”) = g’l” Tr
(i)= 4 gk”
(A.31)
and
(A.33) Now
T r (7’1y” y P ) = T r ( y P y’l 7”) y p y”y p = iPUPX y~ y5 + g”P y’l - gpp y”+ gp” yp.
(A.34) (A.35)
Therefore,
T r (y’l y” 7 P )
= 0 = T r (yP
y’l 7”).
(A.36)
From this, we generalize that trace of the product of odd numbers of y-matrices is zero. Further, we have
~’l~”y~ya+yay~yuy~=2g~ay’ly”-2g”ay~y~+2g’10 (A.37) Therefore,
Trace of y-Matrices
671
Noting that we can write 5
Y
Li 4!
= - &,pup 7,
YP y’ yp,
(A.39)
we have
Tr
(75
(75
7.) = 0
YP 7.)
=o
T r (75 y yv yp)
=0
(A.40) (A.41) (A.42)
and
%) = 42 Epvpu, (A.43) with the definition ~ 0 1 2 3= 1 and E~~~~ = E 2). .k while E~~~~ = -1. In calculations, we usually come across the matrix elements of the form
Tr
Therefore,
( 7 5 Yp Yv Yp
672
Quantum Field Theory [A Summary]
1
-
4
m1m2
X
n
[(h+ m2) 7 p (1 + a75) (h + ml)
(1 + a75)I (A.48)
Here Fz = y - h
F1
=y*h.
(A.49)
Using the formulae for the traces of y- matrices given previously, we get
Similarly for
we get
Spin 1 Field
673
we get the same value as given in Eq. (52).
A.4
Spin 1 Field
Electromagnetic field (photon) with mass m = 0. In the absence of interactions, the electromagnetic field A, (z) satisfies the field equation
@ A , (z)= 0.
(A.54)
There is an additional condition
8, A, (x)= 0.
(A.55)
The Fourier decomposition of A, (x):
(A.56) where E& (x), are four vectors called polarization vectors. ax (k)and u i ( I c ) are interpreted respectively as the annihilation and creation operator of the photon with momentum k and polarization E.; (k). They satisfy the following commutation relations
,
(k’)]
= SA A’
(k),
(k’)]
= [a: (k),
[UX (k) uit [UA
The polarization vector
S3 (k- k’),
(A.57)
~ 1(k’)] , = 0.
(A.58)
(k) satisfies the following relations
E;
EX (Ic)
*
EX’
(k)
=
6,
(A.59) (A.60)
k . 2 = 0. For transverse photon polarization, the four-vector chosen as &; = (0, EX 7
(W
(W)
&&
(k) can be (A.61)
Quantum Field Theory [A Summary]
674
so that we have
k . &x = k *x =~O
(A.62)
and
where q
=
(1,0,0,0).
A.5 Massive Spin 1 Particle A spin 1 particle of mass m is described by a vector field 4,
(z) ,
which in the absence of interactions satisfies the equation (m2
+
0 2 )
4, (x)= 0
(A.63a)
with the subsidiary condition
4, (x)= 0. The Fourier decomposition of dP (z) is given by ap
c 3
x
E$
( k ) [ax ( k ) e-zk.'
(A.63b)
+ b i ( k ) eik.']
X=l
(A.64a)
(A.64b)
Feynman Rules for S-Matrix in Momentum Space
675
ax (Ic) and bx (k) satisfy the following commutation relations:
(k’)]
=
6~
J3 (k- k’),
(A.65a)
[ b (k) ~ ,bi, (k’)]
=
6~ 1’ 63 (k- k’).
(A.65b)
[UX
(k),
u i (k) (UX(k))are creation (annihilation) operators for the particle with polarization X and momentum k. bl (k) (bx (k)) are creation (annihilation) operators for the antiparticle with polarization X and momentum k. The polarization vector E; satisfies the relation EX
*
k*&’=O
EX’ = 6x A’,
(A.66a) (A.66b)
A.6
Feynman Rules for S-Matrix in Momentum Space
For each internal photon line:
-0
For each internal fermion line:
I
For each internal pion line: For each external fermion line entering the graph, depending upon whether the line is in the initial or final state
For each external fermion line leaving the graph, depending upon whether the line is in the final or initial state
(27r) k +la
&a
*------
a
-@
+ * ,
1 k2-m?+ie
-&mT (p) or &P
+*
@T(P)
f i E T (p)
or
&
E ’ T
(PI
676
Quantum Field Theory [A Summary]
For each external photon line: For each external spin 0 meson line: For photon-fermion vertex: 0
For pion-fermion vertex:
I I I
I For photon-meson vertex:
A factor ( 2 ~64) ( ~p - p‘ f k) at each vertex
A factor (-1) for each closed fermion loop For a massive vector boson of mass mw This gives the propagator of the vector boson in unitary gauge. Further one has J d4 1 for each loop integral where the four momentum 1 is not fixed by energy-mometum conservation. Multiply by 6p = l (-1) and -1 (1) respectively for the direct and exchange term of fermion (antifermion)-fermion (antifermion) scattering. Feynman rules for a hermitian self-interacting spin 0 boson with the Lagrangian
are as follows
An Application of Feynman Rules
677
For each external line For each internal line For vertex
A factor (2?r)464(kl+ k2 - k3 - k4) at each vertex For each loop integral J d4Z statistical factors *----
# - - - -
-2!1
, 1 ,
- . - - -'I
+!
\
j etc.
- - - - - I
-----
'd
-4'
A.7
, 6-
An Application of Feynman Rules
As a simple application of Feynman rules, we consider the process e-+e+
+ p-+p+
Pl+P2
= P:+P;
+
Therefore, using the relation S = 1 i ( 2 ~64) (Pi ~ - P f )T :
V
(A.68)
Quantum Field Theory [A Summary]
678
Figure 1 One photon exchange Feynman diagram for the process e- e+ 4 p- p+.
and we put
Therefore,
Using Eqs. (51)-(53) ( u = 0), we get
+
P’, * P2 P; * Pl +Pa * P2 Pi * Pl m: Pi * P; 2 +m, p2 pl 2rn:rnE
=-
+
+
(A.71) Now s = (p1 + p 2 ) 2 = (pi +p;)2 = E
L
= k2
(A.72)
and in the center of mass system p1=
-P2 = P;
Pi = -P2I = PI
(A.73)
An Application of Feynman Rules
679
with IPI =
JG' 2
7
lP'l
=
Jq .
(A.74)
Therefore,
P', P2 Pi PI + P'z P2 P: ' PI *
*
*
-
and
IF1
2
=
e4
S2
+4s (ma
+ m;) (1 - cos28 ) + 16 m:mz
cos2e]
.
(A.76)
Hence from Eq. (2.39), we get
where
Quantum Field Theory [A Summary]
680
(A.78)
(A.79) In the relativistic limit, ,Be z 1 , ,LIP M 1 (s >> rn;,mi) ,we have da dR
- =
a =
A.8
- (l+COS20)
a2 4s
(A.80a)
47ra2 1 3 s'
(A.80b)
--
Charge Conjugation
Dirac equation in the presence of electromagnetic field is given by
(a, + i e A p )- rn] 9 (x)= 0. For the adjoint, field T, Eq. (1) can be written: [-i ( 7 ~(3,) -~ i e A,) - rn] qT (x)= 0. [i?"
(A.81)
(A.82)
Under the charge conjugation
9 (x) A , (x)
-+ +
9,(x)= u,@ (X) Uc-l A; (x)= U,A, (z)UC-'.
(A.83) (A.84)
681
Charge Conjugation
If the Dirac equation is invariant under charge conjugation then:
Now we can write Eq. (83) as
C [-i ( y p ) T (a, - i e A,)
- m]
C-'CqT (x)= 0,
(A.86)
where C is a unitary matrix, called the charge conjugation matrix. Equation (87) is identical to Eq. (86), provided that
y, A;
=
-c ( y q T c-l
=
-A
CL
(A.87) (A.88)
9 I C ( x ) = CTT(x).
(A.89)
c9-T
(A.90)
Also one can write QC(Z) =
9[IC(z)=
(x)= -yocq* -VT(x)C-1.
(A.91)
In Pauli-representation fo y-matrices for p = 0 , 2 for p = 1 , 3 .
(A.92)
Therefore we have from Eq. (88): (A.93) (A.94) Hence in this representation Q" (x)= -2y2Q'.
(A.95)
Quantum Field Theory [A Summary]
682
On the other hand, in the Weyl representation of y-matrices
(A.96) In this representation, one can write
(A.97) where
are two component left-handed and right-handed spinors. In this representation the relations (93) - (96) are again satisfied. Hence from Eq. (96), we get,
(A.99) Sometime it is convenient to write a right-handed field in terms of a left-handed antiparticle field (cf. Eq. (100)): 7 = ig2Jc'
(A. 100)
so that Eq. (98) becomes Q =
(
6 ig2J"*
)
(A.lO1)
Charge Conjugation
683
The Majorana spinor !PM is defined as
Qft
-T
!PM=C!PM
=
= cap%$.
*'Ma
Hence in the Weyl representation
(A.102) We also note that in the Weyl representation:
Y =( y 0
0
)
(A.103)
where f7p
=
(1'2) = (1,Z)
ap
=
(1, -2)
=
(1,- 2 ) .
(A.104)
Now the Dirac Lagrangian
L=
(iypa, - mD)9 + % (QTC-'!P - $ C a T ) 2
(A.105)
(where the second term in Eq. (106) is the Majorana mass term and violates lepton number conservation) can be written in terms of two component chiral fields using Eqs. (99), (loo), (104) and (106):
L =
[ct(Tpap(+ c c ' 8 p a p c c ] - m D [c*T '&g . 2 cc'
$.
tCT ( - i g 2 ) [] (A.106)
where we have used 02
02
z
($y
Quantum Field Theory [A Summary]
684
'
(fermion fields anticommute)
-
-3,("'-P
-
E ~ ' P ~ ~(partial E ~ integration).
c
Equation (107) can be put, in the compact form
where i, j =1, 2 and mij is the symmetric mass matrix
Bibliography
A.9
685
Bibliography
1. J. J. Sakurai, Advanced Quantum Mechanics, Addison- Wesley, Reading, Massachusetts (1967). 2. J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields, McGraw-Hill, New York (1964). 3. C. Itzykson and J. B. Zuber, Quantum Field Theory, McGrawHill, New York (1980). 4. M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, Massachusetts (1995).
Appendix B
RENORMALIZATION GROUP AND RUNNING COUPLING CONSTANT B.l
Feynman Rules for Quantum Chromodynamics
For canonical covariant quantization, the QCD Lagrangian given in Eq. (7.32) is written as [repeated indices imply summation]
1
Tr (TA,T B )= 2
fACD
fBCD
~ A B for ,
= c 2 (G) 6 A B
=
(TA);
the fundamental representation.
(TA); = C F
N
~ A B ,
for S U ( N ) gauge group (B.2)
N2 - 1
6; = 2N ~
-&
b:,
for S U ( N ) .
In the Lagrangian (I), (8, G A , ) ~ is the gauge fixing term, ( being the fixing parameter. The supplementary nonphysical fields, called ghosts are needed for covariant quantization in 687
Renormalization Group and Running Coupling Constant
688
order to cancel the probabilities of observing scalar (or time-like) and longitudinal gluons. Quantizing in a renormalizable gauge leads to the following Feynman rules:
gluon propagator
a
=k
b
s;IcrJ;kLLLC,~
B, p
26; A,quark propagator
Renormalization Group
(i) (ii) (iii)
689
The other factors are J 4 for each loop integral (2r) (-1) for closed fermion (ghost) loop Statistical factors like
B .2 Renormalization Group, Effective Coupling Constant and Asymptotic Freedom We now show that the self-coupling of gluons envisaged in the first term of the Lagrangian (1) has the consequences that QCD has a remarkabIe property of being asymptotically free i.e. the quark - quark force becomes weak at large momentum transfer or short distances, such as probed in deep-inelastic collision [cf. Chap. 141. In other words, the coupling constant a, depends on the momentum transfer in such a way that a, ( Q 2 )--+ 0 as Q2 --f 00. Consider the radiative corrections to quark - quark - gluon (qqG) vertex, where at one loop level these corrections are shown
Renormalization Group and Running Coupling Constant
690
in fig. 1, [Q2= - q 2 ] .
+
+
One loop corrections to quark - quark - gluon (qqG) vertex. The one loop corrections to qqG vertex shown above are infinite. One must define a high Z2- cut off (I being loop momentum) X2, so
691
Renormalization Group
that the loop integrals converge. We have then ~ A = F -2
TA?'p
r s (Q2,
(B.3a)
A, gs)
where
where - .denotes the corrections from higher order loops. Note here that the cut-off dependent logarithmic contributions from diagrams A [involving quark self-energy diagram] and the first of diagrams B [involving the quark gluon vertex function] cancel due to gauge invariance as is also the case in quantum electrodynamics. Since the theory is renormalizable, we must be able to write it as
where p is called the renormalization scale and 2, is a multiplicative renormalization constant. One may define the renormalization scale through the relation
Then neglecting
a0
in Eq. (3b)
and [cf. Eq. (3b)l
rs (Q2/X2, Thus
gs) = g J Y 2 ( X 2 / Q 2 , 9s)
.
(B.6b)
692
Renormalization Group and Running Coupling Constant
This relation expresses the basic-renormalization group property. It is more conveniently expressed through an equivalent differential equation which follows from the p-independence of r3 so that, d r S = 0, (B.8a) dP
or, using Eq. (4), (B.8b) This can be rewritten a s [I?:
( p ) = gs ( p ) ]
(B.9a)
so that, (B.9b) where Z;l2 is given in Eq. (6a). Equations (9) are known as the renormalization group equations for the effective coupling constant, gs ( p ) . Writing
Eq. (9b) gives
=
-29; [bo
+ bl g," + . . .] ,
(B.lob)
where we have used Eq. (6a). To integrate Eq. (9a), it is convenient to write it, on using Eq. (lob), as [putting p2 = Q 2 ]
693
Renormalization Group
or -1
d lnQ2 = d (l/g:)
(l/g?)-'
+ a]
(1/gi)
-1
+
a
*
-
]
.
(B.11)
If we keep only the lowest order term, we have (B.12a) where a, =
g, 6 = 87r bo. Integration of Eq. (12a) gives a;' ( Q ~ )= a;1 ( p 2 ) + bln -. Q2
(B.12b)
P2
Note that what renormalization group does is to relate the coupling constant at two different scales. We may also write Eq. (12b) as Q2
( Q ~=) bln-
(B.12c)
AbCD
where [a;' ( p 2 )= a;']
or (B.12d)
A Q ~ Dis one parameter which determines the size of a, ( Q 2 ) .It must be determined from experiment. Thus finally we have from Eq. (12)
-
1
+0
bln $AQCD
(a: ( Q 2 ) ) .
(B.13)
694
Renormalization Group and Running Coupling Constant
Table B . l Renormalization Constants
Note that we have been able to sum the leading logs here [compare (13) with a p (1 - a,, b In . in the ordinary perturbation theory]. Thus Eq. (13) goes beyond the ordinary perturbation theory. The perturbation now is with respect to 0,(Q2). We now determine b. For this purpose, we need 2, [cf. Eqs. (8) and (9)]. But we note from Fig. 1 that Z,is given by
5+
a)
where the renormalization constants Z ~ F2,1 and ~ 2 3 arise respectively from diagrams A [self-energy part of the fermions (quarks) propagator], B [vertex part for the fermion] and C [the vacuum polarization or the self-energy part of the gluon propagator]. The values of these constants are summarized in Table 1, which also includes Z1 which corresponds to the triple gluon vertex [i.e. the first of diagrams (B) with the quark lines replaced by the gluon
w]. 3
\ lines while the second is replaced by
Cz and
In this table
are defined in Eq. (1) and n.f denote the number of fermion flavors. F’rom Eq. (14) and Table 1, we have to order 9,” CF
Renormalization Group
695
Thus from Eq. (lob) [note that the gauge fixing parameter [ is canceled out] (B.16a) P ( % ) = - a d bo + o (d)]7 so that
(B.16b) (B.16~) Hence in summary, we have from Eq. (13) Q,
1
(Q2) = a-1 P
+ J- (Y c ~- i nr) In $ + 0
(0; ( Q 2 ) )
47r
(B.17a)
(B.17b) It is a very useful equation and the single parameter AQCD becomes the QCD scale which effectively defines the energy scale at which the running coupling constant, attains its maximum. A Q c D can be determined from experiments and turns out, to be [see Chap. 71 A Q ~ D=
140 & 60 MeV.
Note that for SU, (3) [C2 = 31 , (11 -
5
(B.18) nf)is positive for
nj < 11 (which is certainly true for known six quark flavors n,f = 6 ) and then a, ( Q 2 )decreases as Q2 increases. This is made possible because of coefficient, 11 which comes from the self coupling of gluons, non Abelian nature of QCD. The logarithmic deviation from asysmptotic freedom is a characteristic of QCD and the tests of the theory have to be sought to detect, logarithmic scaling violations.
696
B.3
Renormalization Group and Running Coupling Constant
Running Coupling Constant in Quantum Electrodynamics (QED)
For QED, only fermion loops (i.e. the first, of diagrams C in Fig. 1 with gluon replaced by photon and g: by e2) contribute to electric charge renormalization so that, in Table 1 only Z3 without C2 is relevant,. Note, however, that, the contributing charged fermions are e , u, d, p , c, s , T , b and t so that, e2 ( n , f / 2 in ) the expression for Z3 is replaced by
e 2 x Q ; = e 222 f where (n,,/2) are the number of generations [3 in our case] and the factor 3 outside the parenthesis is due to the color. Thus (B.19a) giving (B.19b) The equation analogues to (12) is t,hen (B.20a) giving
(B.20b) and increases with Q2 in cont,rast to a 3 ( Q 2 which ) decreases with Q2.
Let us apply Eq.(20b) for pz = m:, where a,(m;) is determined from Thompson scatkering, for example, [a = ae(rnz)= No matter how small a one has, one can always increase Q2 to a
&]
Running Coupling Constant for SU(2) Gauge Group
697
point where a,(Q2)which was given in Eq.(20b) becomes infinite [Landau ghost]. This, however, occurs [for six flavors] at
Q2 = m2exp (fa-')
GeV2
N"
(B.21)
which is even larger than Ad: M GeV2 by several orders of magnitude . Finally, we wish to remark that the formula (20) holds for m: 5 Q2 < mL. For Q2 2 we have to consider the contribution of charged W* bosons to Pem. In this case
mL,
(B.22)
47r and ae(Q2)still increases with Q2 for n f
=6
(or > 6 ) .
B.4 Running Coupling Constant for SU(2) Gauge Group For SU(2) group from Eq. (2) C2 = 2 and therefore from Eq. (16c) bSW(2) =
(-
1 22 2 - jnf 4lr 3
)
(B.23)
and correspondingly Eq. (17a) becomes
g,
g2 being the coupling constant associated with where a2 = qqW* vertex, W*, W3 being the gauge bosons associated with SU(2) gauge group. Note that for six quark flavors ( n f = 6 ) , > 4)and a2 (Q') is falling with Q2, although at a rate less than a, ( Q 2 )for the SUC (3) group.
(y
Renormalization Group and Running Coupling Constant
698
Renormalization Group Equation and High Q2 Behavior of Green's Function
B.5
Consider now in general a renormalized Green's function (propagator or vertex function or a related quantity) in QCD denoted by rR
(Pirasip>E)=
Z-' ( A 2 / p 2 ' a s ~ E )
( P i , a s O , ~ O )7
(B.25)
where 2 is a multiplicative renormalization factor and I' on the right-hand side knows nothing about p so that $ = 0. This implies that r R satisfies the renormalization group equation 1 drR
-- rR d p
_--1 d Z dP
or
where [cf. Eq. (9) and (lo)]
P
(as)
1 do, 1 = -gs 2 dp 4n
= -P-
P(gs).
(B.26b)
To simplify matters, let us work in the Landau gauge 5 = 0, then
The above equation also determines the high Q2 behavior of r B . To see this, we first note that there is another constraint on I' which
High Q2Behavior of Green’s Function
699
comes from dimensional analysis. Assume we scale all momenta in r~ (pi,0 8 , P ) 7 pi 4 A pi
D is the dimension of the Green’s function (e.g. for inverse gluon propagator r p 2 and we have D = 2). F is a dimensionless function of dimensionless variables. From Euler’s theorem for homogeneous function N
(B.29)
Put t = 1nX and combine the naive scaling equation (29) with the renormalization group equation (27) [which gives the dynamical constraint] to eliminate p& and obtain
Its general solution can be obtained by the method of characteristics. First one solves [cf. Eq. (26b) with t = Inp] (B.31) with the condition 5, (O,a,) = a,. The general solution of Eq. (30) can then be expressed in terms of that of the above differential equation. In this way one obtains t
r R
(A pa,
p ) = A D r R ( pi,8,
( t )>/I) exP [ - 2 s 0 dt’y (8, (t’))]
(B.32) What we learn from this general solution is that the behavior of Green’s functions when all momenta are scaled up is governed by 6, ( t ). Now as already seen in Sec. 2
,8 [(E,( t ) ) 2 ]= - (5, ( t ) ) 2b +
* *
(B.33a)
700
Renormalization Group and Running Coupling Constant
and similarly we can expand
Thus to solve Eq. (31) in the lowest order, we make use of Eq. (33a) and rewrite it as
dt = -
db, 2bb: [ 1 + .. .]
giving b, ( t ) =
-
1
+
as1 2bt
+ 0 (b.”).
(B.34)
Remember that t In X and this is the same functional dependence for b, as before for a, (Q2)in Sec.2. Thus noting that for large t, 6, ( t ) the use of Eqs. (33) and (34) in the first order enable us to write Eq. (32) for large t or X as
- &,
where y or yo is called the anomalous dimension of F R , which can be determined from Eq. (26b). If it, were zero, we would have obtained canonical scaling behavior A D as in the traditional parton [t lnX], we can model [cf. Chap. 141. Noting that 6, ( t ) say from Eq. (35) that the large Q2 behavior of rR (Q2) is N
&
-
High
Q2
Behavior of Green's Function
701
where the second factor can be written as
(B.37)
&
UC2 - in,] [cf. Eq. (16c)l. Thus it, is clear that the with b = [3 renormalizatlon group Tuation has enabled us to sum up terms of the form [a, (Q2)In Q2] whereas in ordinary perturbation theory we would have to deal with a power series in a, (Q2)ln Q 2 . Analogous logarithmic violation of the scaling will hold in the deep inelastic structure functions and similar physical quantities. Let us now consider some simple applications:
B. 5.1 Gluon propagator From Table 1,
13 [(-[) C, 8?r 3
2a,
2, = 1 + -
^/v =
2[(Y
where
d (-k2)
I,.
+0
c 2-
3 O f ].
4
- t) CZ - 5
x o =8n [ 1
+ O (a:)
-
(Y
-
-E)
4
[a, ( - k 2 ) ] "lVo/b .
(B.38)
(a:) (B.39)
(B.40b)
Renormalization Group and Running Coupling Constant
702
P
P
Figure 1 Fermion self-energy at one-loop level.
B.5.2 Fermion propagator
s,'
( P ) = (r' - m, -
where
c
( p ) = m,
c,
(P2)
c c,(P2)
(B.41a)
(P))1
+ (d - m,)
.
(B.4l b )
Let us define an effective or running mass through the following (B.40~) sc
( P 2 ) = (1 +
c,
(p2))-l
(B.40d)
The fermion self-energy diagram is given in Fig. 2 below. This determines C ( p ) at one-loop level. The renormalization mass mR is defined by m R =
Zm
mB,
(B.42)
where 2, is the multiplicative mass renormalization constant and is given by
High Q2Behavior of Green’s Function
703
(B.43) while from Table 1: Z3F
-’
=
(l-X2)
=
1 - 2-QS ( C F <)ln-. x 4T CL
(B.44)
Thus to the leading order 3;no =
3 CF 47r
(B.45) (B.46)
where for SV, (3) CF =
4 [cf. Eq.(2)]. Hence (B.47a)
while
S F (p2)
1 - [as (-p2)]7Fo’b.
d
For large p2 we note from Eq. (36) that
(B.47b)
704
Renormalization Group and Running Coupling Constant
B.6 Bibliography See bibliography at the end of Appendix A and Sec. C of the bibliography at the end of Chap. 7.
Index A heavy (charmed and botAbelian gauge transformation, tom), 267 magnetic moment of, 192 228 Beauty flavor, 265 Aharanov and Bohm Effect, 221 ,&decay, 47, 380 Aharanov and Bohm Experift values, 50 ment, 220 double, 322 Asymmetry parameter for meaFermi theory, 425 sure of CP-violation, 522 Bosons Asymptotic freedom decay widths of W and 2, evidence for running of a8(Q2), 45 1 231, 233, 234 Bottom quark, 265 property of QCD, 21, 230, Bottomonium, 268 231, 280, 421, 689 states, 276 Axial anomaly, 418, 475 cancellation in electroweak C theory, 479 C-parity, 268 cancellation in gauge t h e Cabibbo angle, 367, 382 ory, 476 Charge conjugation, 120, 513, giving rise to no --+ 27 de680 cay, 214, 418 conservation of, 247 in QCD, 420 in hadronic interactions, in QED, 418 122 invariance, 121, 123, 170 matrix, 613, 681 of J/+, 261 parity, 121 Charge radius
B Baryon decays, 377 Baryon states, 146 Baryons, 8, 10, 143
705
706
mean square, 484 Charges axial vector, 409 baryon, 103 color, 213 electric, 102 electromagnetic, 4 gravitational, 2 lepton, 103 strong color, 11 vector, 409 weak color, 13 Charm, 259, 262 decays of, 275 discovery of, 259 flavor, 262 isospin, 261 spin-parity, 261 SU(3) classification of states, 26 1 Charmonium states, 268 decays of, 275 Chiral SU(2) group, 409 Chiral symmetry, 401, 405 application to non-leptonic decays of hyperons, 416 current algebra, 408 explicit breaking of, 412 CKM matrix, 472, 530, 567 experimental values of matrix elements, 475 Maiani-Wolfenstein way, 530 unitarity triangle, 530 Wolfenstein parameterization, 530
INDEX
Color, 10, 11, 19-21 charges, 213 confining potential, 21, 213, 239 confinement, 12, 213, 238, 239, 287 electric field, 293 electric potential, 19, 235 evidence for, 213 magnetic coupling, 304 magnetic moments, 243, 244 Conservation of baryon charge, 102 electric charge, 102 energy momentum, 27 hypercharge, 104, 105 lepton charge, 103 muon number, 104 parity, 70 probability, 35 strangeness, 104 Conserved vector current hypothesis of, 402 Cornell potential, 242 Cosmology, 21, 612, 623 and particle physics, 623 baryogenesis, 652 baryon asymmetry, 652, 655 baryon density, 639 baryon density parameter, 64 1 baryon energy density, 641 baryon number density,
INDEX
641 bottleneck, 650 CMBR, 632 cosmic history, 640 cosmological constant, 626 cosmological principle, 623 critical density, 627 deceleration parameter, 628 density of universe, 626 density parameter, 627 effective cosmological constant, 658 entropy density, 652 essential features of the standard model, 631 flatness problem, 657 freeze out, 642 horizon problem, 655 Hubble expansion, 631 Hubble parameter, 631 ignorance factor, 631 inflation, 655 inflationary universe, 655, 658 isotropic pressure, 626 neutrino mass, 646 nucleosynthesis temperature, 650 particle density, 627 photon number density, 324 primordial nucleosynthesis, 649 radiation density, 637 radiation era, 635 Riemann zeta function, 633
707
Robert son-Walker metric, 623 standard model, 626 thermal equilibrium, 632 universe; closed, flat and open, 623 CPT theorem, 473 CP-violation, 513, 524, 547, 548 BOBo Mixing and, 527 and Particle Mixing, 513 general formalism, 515 in Bo decays, 533 in K°Ko system, 542-544 in Hyperon Non-Leptonic Decays, 552 in the Standard Model, 525 Cross-section hadronic, 60 Mott, 483 Current algebra and chiral symmetry, 408 Current quark masses, 200,410 Currents axial vector and its partial conservation (PCAC), 405 conserved (Noether's theorem), 228 vector, 406 CVC, 402
D Dalitz plot, 45 Decay widths
INDEX
708
of W and 2 bosons, 451 Deep inelastic scattering, 483 lepton-nucleon, 234, 483, 485 neutrino-nucleon, 498 Bjorken scaling, 510 electron (muon), 483 form factors, 484 involving neutral weak current,s, 509 parity violating, 450 parton model, 493 polarized asymmetry, 489 proton-spin crisis, 508 Sachs form factors, 490 structure function, 490, 494, 510 slim rules, 501 Detailed balance principle, 79 Dirac equation, 665 Dirac gamma matrices, 665,666 Dirac Lagrangian, 683 Duality, 617 String Theory and , 617
E Electromagnetic Interaction, 52 Electroweak unification, 12, 17, 425, 433 pparameter , 438 VV and 2 bosons, 451 energy scale, 18 experimental consequences of the, 441 gauge group, 427
g a w e group [SUL(2) x U Y (1)J , 434 gauge transformation, 427 Higgs-Kibble mechanism, 431 Lagrangian, 438 Lagrangian for charged and neutral currents, 441 R-gauge, 432 radiative corrections, 443 renormalizability , 432 spontaneous breaking, 429 spontaneous gauge symmetry breaking, 426 standard model, 12 weak mixing angle , 13 Equation of continuity, 219 Euler’s theorem, 699 Exchange potential One gluon, 234 (Exclusive) Semi-leptonic decays of D and B mesons, 575
F Fermi constant, 3, 17, 375 Fermi-Dirac statistics, 668 Fermion, 11 generation, 454 Lagrangian, 438 longitudinal polarization, 457 mass matrix, 474 masses, 438 propagator, 702
INDEX
self energy diagram, 702 Fermion flavor, 694 Fermion masses in the standard model, 328 Fermi plot, 48 Feynman rules, 676 application to e+e- --f ,u+,LL-, 677 for QCD, 687 Fierz reordering theorem, 362 Fifth quark flavor, 265 Flavor eigenstates, 347 Flavor symmetry, 299 Form factors weak decays, 575,576,580 in heavy quark limit, 579581 Fundamental forces, 1
G y-matrices, 665, 670 Pauli representation, 666, 681 trace of, 670 Weyl representation, 682 G-Parity, 125 Gauge Abelian, 228 bosons, 224, 458 coupling constants, 604 for electromagnetic interaction, 220 for QCD, 223 force, 12, 216 group, 427
709
group SUc(3), 21 group SUL (2) x U( I), 434 hierarchy problem, 23 invariant Lagrangian, 218, 223, 428 non-Abelian, 228 principle, 213, 218 symmetry breaking, 426 transformation, 102 transformation UQ(l),102 unitary, 432 vector bosons, 15 Gauge principle, 218 in electromagnetism, 220 Gauge symmetry spontaneous breaking, - 15 Gauge vector bosons, 15 Gell-Mann-Nishijima relation, 108, 139 Gell-Mann-Okubo mass formulae, 172 Ghosts, 687 GIM mechanism, 468 Globally conserved quantum numbers , 97 baryon charge, 102 electric charge, 102 lepton charge, 103 muon number, 104 selection rules, 97 Gluon, 20 exchange potential, 20, 21 Gluons, 228 longitudinal, 688 Goldberger-Treiman (G-T) re-
710
lation , 406 Goldstone-Nambu theorem, 430 Grand unification, 22, 601 mass scale, 604 proton decay, 607 Gravitational force, 2 Green's function in QCD, 698 GUTS, 606 General consequences of, 606
H Hadron Spectroscopy, 234 Hadronic cross-section, 60 Hadronic decay width, 278 Hadrons string picture of, 238 Heavy Baryons, 267 Heavy baryons mass formulae for, 307 Heavy Flavors, 259 Heavy flavors weak decays of, 559, 567 Heavy quark color magnetic moment of, 305 propagator in QCD, 297 spin symmetry of, 298 Helicity of the neutrino, 363 Higgs boson bounds on mass, 463
INDEX
coupling, 465 decays, 467 doublet, 606 field, 17 mass, 18, 447, 463 particle, 18, 431 searches, 18, 465 upper bound, 463 Higgs boson mass, 463 unitarity, 463 Higgs field, 15 Higgs particle, 17 Higgsino, 613 HQET, 293 effective lagrangian of, 293 mass spectroscopy for hadrons and applications, 303 Hypercharge, 105 Hyperons chiral Symmetry to nonleptonic decays of, 416 non-leptonic decays of, 386 Hyperons decay A I = 1/2 rule for, 389 '
I Interaction Spin-spin, 243 Internal Symmetries, 97 charge conjugation, 120 G-Parity, 125 Isospin, 106 selection rules, 97 Invariance principle, 65
INDEX
Interaction picture, 31 by a unitary transformation, 32 Isospin, 106 electromagnetic interaction and, 110 weak interaction and, 111
J
Jl+,259 family, 259 spin-parity, 261
K Kaon decay constant, 383 Kaon threebody semi-leptonic decay, 384 Kurie plot, 48
L Lagrangian density for electromagnetic field, 220 for free quarks, 225 Lee-Sugawara relation, 393 LEP, 18, 446, 456, 591, 651 Lepto-quark, 23 Lepton, 8 Leptonic decays of 7 lepton, 559 of D and B mesons, 569
M Magnetic moment neutrino, 355 Magnetic moments of baryons, 192
711
Magnetic moments of quarks, 195 Magnetic moments transitions, 341 Mass spectroscopy for hadrons and applications, 303 Mass spectrum, 243 Mass spectrum of P-wave heavy mesons, 309 baryon, 249 hadrons with one heavy quark, 303 Mesons, 8, 141 bottom, 265 charmed, 263 decays of P-wave heavy, 315 massless pseudoscalar, 405 SU(6) wave function for, 187 Top, 266 Mikheyev-Smirnov-Wolfenstein [MSW] Effect, 343 M-theory, 617 p-decay, 372 Muon decay of polarized, 375
N Nambu-Goldstone bosons, 405, 42 1 Neutral weak interaction, 13 Neutrino, 319 helicity of the, 363 mass, 320
712
Neutrino flavor, 334 Neutrino magnetic moment, 355 Neutrino mass, 320 astrophysical constraints on, 323 constraints on, 321 Dirac and Majorana, 325 seesaw mechanism, 330 Neutrino oscillations, 331 evidence for, 334 resonant, matter, 343 vacuum, 341 Neutron P-decay, 319 half-life, 651 Non-leptonic decays of hyperons, 371, 395 Non-leptonic decays of hyperons, 371, 386 A 1 = 1/2 rule, 389 current algebra approach, 394 in non-relativistic quark model, 393 Lee-Sugawara relation, 393 octet dominance, 393 Non-leptonic decays of D and B mesons, 584 Noether’s theorem, 229 Nucleon magneton, 199
0 Octet, 401 Optical theorem, 82
INDEX
Oscillations KO - KO,542 021 rule, 177
P Parity, 67 intrinsic, 69 Partial wave unitari ty two-particle, 83 Partially conserved axial vector current, 405 Particle Mixing C P -violation, 5 13 Parton model, 493 striicture function, 496 PCAC, 405 consequences of, 405 hypothesis, 405 Phase space, 38 density, 45 spectrum, 113 three-body, 43 two-body Lorentz invariant, 83 Photon, 4-6, 12, 13, 19 exchange, 19 quantum of electromagnetic field, 1 Pion decay, 382 exchange, 60 intrinsic parity, 72 soft, 410 spin, 72, 79 Pion decay constant, 214, 563
INDEX
Planck length, 614 mass, 2, 626 scale, 464 Polarized asymmetry deep inelastic scattering, 489 Potential Cornell, 242 Proton decay, 22 Pseudoscalar meson decays, 382 Positronium, 124, 278
713
vertices, 230 Quantum electrodynamics (QED), 20, 696 running coupling constant, 696 Quantum field theory, 1, 663 application of Feynman rules, 677 charge conjugation, 680 Feynman rules, 676 massive spin 1, 674 spin 0, 663 spin 1, 673 spin 1/2, 665 Quark, 18-21, 185 Q2evolution of gauge coupling confinement, 20, 21 constants, 604 flavor, 9, 11, 19, 185, 213, Quantum chromodynamics (QCD), 216, 244, 262 20, 225 masses (constituent), 200 asymptotic freedom, 20,21, masses (current), 200,410, 231, 421, 687 416 confining long range potenQuark model tial, 21 prediction for baryon magconserved currents, 228 netic moments, 200 effective coupling constant, prediction for radiative de20 cays of vector mesons, Feynman rules , 687 200 gauge invariant couplings, spectator, 591 230 SU(6) and, 185 Lagrangian, 227, 687 Quark-quark-gluon vertex, 232 long range potential, 20 Quarkonium, 268 one gluon exchange potenhadronic decay width of, tial, 238 278 scale factor, 233, 693 leptonic decay width of, sum rules, 421, 501 276
4
INDEX
714
level spacing, 285 mass spect,rum, 285 non-relativistic treatment of, 280
R R-gauge, 432 Radiative corrections need for, 443 Radiative decays of vector mesons, 200 Regge tragectories, 240 connection with confining potential, 240 Relativistic quantum mechanics gauge principle for, 223 Renormalizability , 432 Renormalization factor multiplicative, 696 Renormalization group, 22,605, 687 asymptotic freedom of QCD, 689 effective conpling constant, 689 equation of Higgs boson coupling, 467 high Q2 behavior of Green’s function, 698 renormalization constants, 694 renormalization scale , 691 running coupling constant, 687
running coupling constant for SU(2) gauge group, 697 running coupling constant in QCD, 687 running coupling constant in QED, 696 running mass, 702 Resonance A (delta), 113 p (rho), 113 production, 111 Running coupling constant for SU(2) gauge group, 697 in QCD, 687 in QED, 696
S Scattering process in the center of mass frame, 29 in the laboratory frame, 28 kinematics of a, 27 neutrino-electron , 58 nucleon-nucleon, 60 three-body decay, 43 two-body, 41 See-saw mechanism, 328 Selection rules globally conserved quantum numbers, 97 internal symmetries, 97 Semi detailed balance principle, 79 Semi-leptonic decays
INDEX
of D and B mesons, 570 Sixth quark flavor, 266 S-Matrix, 33, 35 parity constraints on, 73 unitarity of the, 80 Solar neutrinos, 338 Space-time symmetries, 65 Spectrum the mass, 243 Spin of A, 116 of pion, 79 Spinor Majorana, 613, 683 Weyl, 682 Spin projection operators, 669 Spin-spin interaction, 251 Spontaneous gauge symmetry breaking, 426 Spontaneously broken chiral symmetry, 411 Standard model cosmology, 626 electroweak unification, 12 flatness problem, 657 horizon problem, 655 strong force, 12 Standard solar model prediction of neutrino fluxes, 340 Strangeness, 104 String picture of hadrons, 238 String Theory, 617 and duality, 617 heterotic, 618
715
Strings, 601 supersymmetry and, 612 Strong quark-quark force, 19 Structure function parton model, 497 SU(3), 131 invariant BBP couplings, 167 irreducible representation of, 151 mass splitting in flavor, 170 particle representations, 181 sextet representation of, 267 VPP couplings, 168 weight diagrams, 140, 141, 143, 149 Young’s tableaux, 153 SU(3)xSU(3) chiral algebra, 412 SUc(3), 213, 225, 693 SU(6), 185 and Quark Model, 185 SU(N), 159 Young’s tableaux, 159 Sum rules, 501 Adler, 502 Bjorken, 505 Ellis-Jaffe, 507 Gottfried, 503 Gross-Llewellyn Smith, 502 QCD, 421 spin dependent, 503 Super Yang-Mills, 616 Supergravity, 619
716
Supersymmetric Yang-Mills, 615 Supersymmetry, 601, 613 and strings, 612 Symmetries internal, 97 parity, 67 space-time, 65 time reversal, 76
T r-lepton leptonic decays of, 559 semi-hadronic decays of, 562 T-duality, 618 Time reversal, 76 T-Matrix, 37, 382 unit ar it,y constraints, 80 Top Quark, 266 mass and width, 266 Trace of y-matrices, 670 Trace techniques, 373,375, 485, 572 Transition matrix, 37 Transition rate, 31
U U-spin, 151 Unification electroweak, 17, 425 grand, 21, 601 Unitarity constraints, 80 on Fermi theory, 425 on I&, 530
INDEX
partial wave, 85, 89 Unitary Group, 131 Unitarity triangle, 530 Units, 24 Upsilon (T)family, 273 spectrum, 274
V V - A interactions, 361 vector mesons Radiative decays of, 200 Virial theorem, 282
W Weak decays of heavy flavors, 559 Weak Decays of Heavy Flavors, 567 Weak hadronic currents properties of, 401 Weak interactions, 57, 361 cross-section, 59 neutral, 441 scale, 440 V-A, 362 Weak isospin, 13, 433 Weak mixing angle, 13, 438, 44 1 experimental determinrl, tion, 443, 448 radiative corrections, 445 Weak processes, 364 non-leptonic, 370 purely leptonic, 364 semi-leptonic, 364, 376
INDEX
Weak vector bosons, 15, 364 decay width, 451 exchange graph, 393 forward-backward asymmetry, 454 longitudinal polarization, 457 masses, 18, 437, 441 production, 460, 461 Yang-Mills character, 458 Wilson coefficients, 586 Wu-Yang convention, 545
Y Yang-Mills fields, 228 Young’s tableaux, 153 for SU(N), 159
717
